INELASTIC WEB CRUSHING PERFORMANCE LIMITS OF
HIGH-STRENGTH-CONCRETE STRUCTURAL WALLS

By
Xuejian Liu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
CIVIL ENGINEERING
2010

ABSTRACT
INELASTIC WEB CRUSHING PERFORMANCE LIMITS OF
HIGH-STRENGTH-CONCRETE STRUCTURAL WALLS
By
Xuejian Liu
The construction of the three new toll bridges in California has highlighted the
application of the hollow rectangular bridge piers with highly-reinforced boundary elements
and the connecting walls. However, current conservative design approaches result in massive
cross sections that can lead to increased construction cost. This study investigates the
feasibility of defining new limits for a seldomly considered failure mode that could lead to
more slender cross sections with the use of high-strength-concrete (HSC).
The possibility of web crushing failure in shear dominated reinforced concrete
element rises when the dimensions of the boundary elements become larger and the webs
becomes thinner. Web crushing, or diagonal compression failure, manifests itself as
compression failure of concrete struts formed by diagonal tension cracking in the wall web.
The failure is brittle in nature and therefore, it is mandated to be suppressed for design in
seismic regions. However, it is hypothesized in this study that web crushing failures may
occur after significant and stable ductile response by using HSC by increasing the capacity to
crushing capacity proportionally to the concrete‟s compressive strength. To verify the
hypothesis, eight 1/5-scale cantilevered structural walls with highly confined boundary
elements were tested with design concrete compressive strengths of 34, 69, 103, and 138
MPa (5, 10, 15 and 20 ksi) under cyclic and monotonic loading protocols. The experimental
results revealed that HSC can effectively delay web-crushing failures and increase the
displacement ductility levels of structural walls.

To evaluate the seismic performance of HSC hollow bridge piers, two 1/4-scale
hollow pier units, consisting of an assembly of four walls with heavily confined corner
elements, were subjected to simulated seismic demands in diagonal and multi-directional
directions with design concrete strengths of 34 and 138 MPa (5 and 20 ksi), respectively.
Both test units exhibited stable ductile behavior until web crushing at moderate ductility
levels. The comparable ductility capacities of the pier test units indicate that the
advantageous effect of HSC in improving web crushing capacity is compromised by
concrete damage and strength degradation under multi-directional loading.
Comprehensive nonlinear finite element (FE) modeling was conducted through 3D
FE analyses with plasticity-based constitutive models and 2D FE analyses with
phenomenological constitutive models to enhance understanding of web crushing capacity in
structural walls. The results were compared with test data at the global and local levels to
evaluate the performance of analytical methods in predicting flexural and shear behavior.
The analytical models revealed that the concrete stress demand at web crushing is only a
small portion of the cylinder concrete compressive strength due to the complex stress state
in the web crushing region. The results also confirmed that web crushing, in the form of an
inelastic shear failure, is caused by flexure-shear interaction effects in the wall web.
In view of the complexity and enormous efforts entailed in finite element modeling
as well as the inability of the modeling approaches in capturing failure, simplified analytical
method based on truss models extracted from observations of the inelastic shear cracking
and failure mechanisms were implemented. This was done by modifying an existing
comprehensive inelastic strut-and-tie model for assessment of web crushing capacity.
Calibration of the model with experimental results showed that the modifications led to
improved predictions of web crushing capacity of high-strength-concrete structural walls.

To my wife Tingyao and my son Cephas.

iv

ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor, Dr. Rigoberto Burgueño
for his continuous support and encouragement. His enthusiasm and devotion to educate
students on how to become a researcher impressed me a lot. I am very thankful to him for
his understanding, trust and for never giving up on me, which forged me through the
enduring PhD study. It has been a challenging and fruitful experience to work under his
supervision.
I wish to thank the members of PhD committee, Dr. Ronald S. Harichandran, Dr.
Eric M. Hines, Dr. Jung-Wuk Hong, and Dr. Dahsin Liu for their expertise and precious
time. Special thanks are due to Dr. Eric M. Hines for his encouragement. He sets an example
to me on what an excellent structural engineer should be. His research publications have
been textbooks for my study.
I would like to thank Mr. Siavosh Ravanbaksh for his help at the Civil Infrastructure
Laboratory. His warm welcome and detailed tutorship made my time at the lab enjoyable.
Lastly, and most importantly, I would like to thank my family. To my wife Tingyao, I
cannot image what my life will be without you. To my son, Dad learns gradually on how to
be a Dad on your behalf. I am grateful to all of our family members. I know I am not alone
when I think about you.
This study was mainly funded by the National Science Foundation under grant NSFCMS-0530634. Additionally, financial support in the form of assistantships and fellowship
awards obtained from the Department of Civil and Environmental Engineering, and the
Graduate School at Michigan State University are gratefully acknowledged. Special thanks are
addressed to late Dr. Delia Koo for the scholarship granted by the International Studies and
Programs at Michigan State University.
v

TABLE OF CONTENTS
LIST OF TABLES ...................................................................................................... IX
LIST OF FIGURES ...................................................................................................... X
CHAPTER 1.
INTRODUCTION ........................................................................................................ 1
1.1 Motivation .................................................................................................................................... 1
1.2 High-strength-concrete in seismic regions .............................................................................. 5
1.3 In-plane Web Crushing Behavior ............................................................................................. 6
1.4 Diagonal Web Crushing Behavior ............................................................................................ 8
1.5 In-plane Cyclic Loading Effect ................................................................................................. 9
1.6 Multi-directional Loading Effect ............................................................................................ 10
1.7 Objectives ................................................................................................................................... 11
1.8 Approach .................................................................................................................................... 12
1.8.1 Experimental studies on single structural walls ...........................................................12
1.8.2 Experimental studies on wall assemblies ......................................................................12
1.8.3 Nonlinear finite element modeling ................................................................................13
1.8.4 Simplified analytical modeling ........................................................................................13
1.9 Scope ........................................................................................................................................... 14
1.10 Organization .............................................................................................................................. 14
CHAPTER 2.
LITERATURE REVIEW AND STATE-OF-THE-ART ........................................... 17
2.1 Summary ..................................................................................................................................... 17
2.2 Force-Displacement Behavior of Structural Walls ............................................................... 17
2.3 Historical Overview of Shear Strength Models .................................................................... 19
2.4 US Code Provisions for Shear................................................................................................. 20
2.4.1 ACI-318-08 Shear Provisions .........................................................................................20
2.4.2 AASHTO Standard Specifications for Highway Bridges ...........................................21
2.4.3 AASHTO LRFD Bridge Design Specifications ..........................................................21
2.4.4 Discussion .........................................................................................................................22
2.5 Web Crushing Capacity Models .............................................................................................. 25
2.5.1 Oesterle et al. ....................................................................................................................25
2.5.2 Paulay and Priestley .........................................................................................................27
2.5.3 Hines and Seible ...............................................................................................................29
2.5.4 Discussion on Truss Models ..........................................................................................32
2.6 Nonlinear Finite Element Modeling Method ....................................................................... 33
2.6.1 Continuum Damage Plasticity model in ABAQUS ....................................................33
2.6.2 MCFT based FE modeling – VecTor2 .........................................................................41
2.7 Research Hypothesis of This Study ........................................................................................ 45
CHAPTER 3.
EXPERIMENTAL STUDIES ON HSC STRUCTURAL WALLS............................ 48
3.1 Summary ..................................................................................................................................... 48
vi

3.2 Experimental Investigation ...................................................................................................... 49
3.2.1 Single Wall Test Units .....................................................................................................50
3.2.2 Material Properties ...........................................................................................................54
3.2.3 Instrumentation ................................................................................................................57
3.2.4 Loading protocol ..............................................................................................................57
3.3 Experimental Observations ..................................................................................................... 60
3.4 Experimental Results ................................................................................................................ 75
3.5 Discussion .................................................................................................................................. 89
3.6 Conclusions ................................................................................................................................ 92
CHAPTER 4.
EXPERIMENTAL STUDIES ON HSC WALL ASSEMBLIES UNDER MULTIDIRECTIONAL LOADING ...................................................................................... 94
4.1 Summary ..................................................................................................................................... 94
4.2 Experimental Investigation ...................................................................................................... 95
4.2.1 Wall-Assembly Test Units...............................................................................................95
4.2.2 Materials ............................................................................................................................98
4.2.3 Test setup ....................................................................................................................... 100
4.2.4 Instrumentation ............................................................................................................. 102
4.2.5 Loading Protocol........................................................................................................... 105
4.3 Experimental Results .............................................................................................................. 107
4.4 Discussion ................................................................................................................................ 121
4.5 Conclusions .............................................................................................................................. 129
CHAPTER 5.
NONLINEAR FINITE ELEMENT MODELING OF HSC STRUCTURAL
WALLS………… ......................................................................................................... 131
5.1 Summary ................................................................................................................................... 131
5.2 Introduction ............................................................................................................................. 132
5.3 ABAQUS Analysis of Single Walls under Monotonic Loading ....................................... 135
5.3.1 Geometrical Modeling .................................................................................................. 135
5.3.2 Material Properties ........................................................................................................ 138
5.3.3 Solution Controls .......................................................................................................... 146
5.3.4 Results ............................................................................................................................. 146
5.4 VecTor2 Analysis of Single Walls under Monotonic Loading ......................................... 157
5.4.1 Geometrical Modeling .................................................................................................. 157
5.4.2 Material properties ........................................................................................................ 158
5.4.3 Solution Controls .......................................................................................................... 159
5.4.4 Results ............................................................................................................................. 159
5.5 ABAQUS Analysis of Single Walls under Cyclic Loading ................................................ 167
5.6 VecTor2 Analysis of Single Walls under Cyclic Loading .................................................. 167
5.6.1 General description ....................................................................................................... 167
5.6.2 Results ............................................................................................................................. 167
5.7 Discussion ................................................................................................................................ 174
5.8 Conclusions .............................................................................................................................. 175
CHAPTER 6.
SIMPLIFIED ANALYTICAL METHOD ON HSC STRUCTURAL WALLS ....... 178
vii

6.1
6.2
6.3
6.4
6.5

Summary ................................................................................................................................... 178
Concrete softening parameter ............................................................................................... 178
Results ....................................................................................................................................... 183
Discussion ................................................................................................................................ 186
Conclusions .............................................................................................................................. 190

CHAPTER 7.
MODELING AND ANALYSIS OF HSC HOLLOW BRIDGE PIERS ................... 191
7.1 Summary ................................................................................................................................... 191
7.2 Diagonal Web Crushing Capacity ......................................................................................... 191
7.3 NLFEM of the Diagonal Pier Test Unit ............................................................................. 193
7.4 Equivalent Single Wall Unit Analysis ................................................................................... 199
7.5 Conclusions .............................................................................................................................. 204
CHAPTER 8.
SUMMARY AND CONCLUSIONS ......................................................................... 205
8.1 Summary ................................................................................................................................... 205
8.2 Original Contribution ............................................................................................................. 208
8.3 Future Research Needs .......................................................................................................... 209
8.4 Research Impact ...................................................................................................................... 210
REFERENCES. ......................................................................................................... 211

viii

LIST OF TABLES
Table 3-1 Summary of the concrete mixtures for the single wall test units...............................54
Table 3-2 Concrete material properties at day-of-test: Compressive strength, tensile strength
and modulus of rupture for the single wall test units. .........................................................55
Table 3-3 Steel properties and modeling parameters for the single wall test units. .................56
Table 3-4 Values of force at first yield and ideal yield displacement for single wall tests. ......60
Table 3-5 Single wall tests: flexural and shear deformations versus ductility levels and
comparison of the calculated and measured global displacement .....................................79
Table 3-6 Comparison of inelastic web-crushing capacities of single walls with shear models.
......................................................................................................................................................91
Table 4-1 Concrete mixtures for pier test units. ............................................................................99
Table 4-2 Steel properties and modeling parameters for pier test units.................................. 100
Table 4-3 Values of force at first yield and ideal yield displacement. ...................................... 107
Table 4-4 DPT unit: peak force-displacement values. ............................................................... 115
Table 4-5 BPT unit: peak force-displacement values. ................................................................ 117
Table 5-1 Fracture energy of concrete materials ........................................................................ 140
Table 5-2 The concrete and steel material parameters for M05M. .......................................... 142
Table 5-3 The concrete and steel material parameters for M10M. .......................................... 143
Table 5-4 The concrete and steel material parameters for M15M. .......................................... 144
Table 5-5 The concrete and steel material parameters for M20M. .......................................... 145
Table 5-6 Summary of the material behavior models in VecTor2. .......................................... 158
Table 6-1 Comparison of inelastic web-crushing capacities of single walls under cyclic
loading with Modified Hines-Seible model. ....................................................................... 186
Table 7-1 The concrete and steel material parameters for DPT Unit. .................................... 195

ix

LIST OF FIGURES
Images in this dissertation are presented in color.
Figure 1.1 SFOBB pier test unit: Present design (Left), Potential design (Right). ..................... 2
Figure 1.2 Flexure-shear and elastic shear mechanisms: (a) In test wall, (b) Sketch of the
regions, (c) Truss models. .......................................................................................................... 8
Figure 2.1 Analytical force-displacement responses of the single walls with ACI-318 shear
stress limits. ................................................................................................................................24
Figure 2.2 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Oesterle et al. .............................................................................................27
Figure 2.3 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Paulay et al. ................................................................................................29
Figure 2.4 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Hines and Seible........................................................................................32
Figure 2.5 Concrete stress-strain curve under uniaxial tension ...................................................38
Figure 2.6 Concrete stress-strain curve under uniaxial compression .........................................38
Figure 2.7 Illustration of the effect of the compression stiffness recovery parameter

. ....39

Figure 2.8 The modified compression field theory (MCFT) [40]. ..............................................42
Figure 3.1 Single wall test unit cross sections with reinforcement details. ................................52
Figure 3.2 Single wall test setup overview. .....................................................................................53
Figure 3.3 Concrete material strength development curves for single wall tests. .....................55
Figure 3.4 Single wall test unit instrumentation layout. ................................................................59
Figure 3.5 Cyclic loading protocol for single wall tests. ...............................................................60
Figure 3.6 Test Unit M05C: south face,

= 1 x +1, push (positive) towards west. ..............64

Figure 3.7 Test Unit M20C: south face,

= 1 x +1, push (positive) towards west. ..............65

Figure 3.8 Test Unit M05M: south face, force control
Figure 3.9 Test Unit M05C: north face,

, push (positive) towards west. ...66

= 1.8 x +1, push (positive) towards west. ...........67
x

Figure 3.10 Test Unit M10C: north face,

= 2 x +1, push (positive) towards west. ...........68

Figure 3.11 Test Unit M15C: north face,

= 4 x +2, push (positive) towards west. ............69

Figure 3.12 Test Unit M20C: north face,

= 6 x 1, push (positive) towards west. ...............70

Figure 3.13 Test Unit M05M: north face,

= 2.3, push (positive) towards west. ..................71

Figure 3.14 Test Unit M10M: north face,

= 7, push towards west. ......................................72

Figure 3.15 Test Unit M15M: north face,

= 6.5, push towards west. ...................................73

Figure 3.16 Test Unit M20M: south face,

= 9.2, push towards west. ...................................74

Figure 3.17 Test Unit M05C: Hysteretic loops. .............................................................................77
Figure 3.18 Test Unit M10C: Hysteretic loops. .............................................................................77
Figure 3.19 Test Unit M15C: Hysteretic loops. .............................................................................78
Figure 3.20 Test Unit M20C: Hysteretic loops. .............................................................................78
Figure 3.21 Test Unit M05C: Hysteretic loop in flexure. .............................................................80
Figure 3.22 Test Unit M05C: Hysteretic loop in shear. ................................................................80
Figure 3.23 Test Unit M20C: Hysteretic loop in flexure. .............................................................81
Figure 3.24 Test Unit M20C: Hysteretic loop in shear. ................................................................81
Figure 3.25 Test Unit M20C, shear displacements as a function of the flexural displacements
......................................................................................................................................................82
Figure 3.26 Test Unit M05C: Comparison of the shear deformation between the bottom and
top panel. ....................................................................................................................................82
Figure 3.27 Test Unit M20C: Comparison of the shear deformation between the bottom and
top panel. ....................................................................................................................................83
Figure 3.28 Test Units M10: Curvature profiles (average of push and pull maximums). ........84
Figure 3.29 Test Units M20: Curvature profiles (average of push and pull maximums). ........84
Figure 3.30 Test Units M10C and M20C: Strain profiles of transverse steel. ...........................85
Figure 3.31 Test Units M05: Comparison of the force-displacement envelopes .....................87
Figure 3.32 Test Units M10: Comparison of the force-displacement envelopes .....................87
xi

Figure 3.33 Test Units M15: Comparison of the force-displacement envelopes .....................88
Figure 3.34 Test Units M20: Comparison of the force-displacement envelopes .....................88
Figure 3.35 Concrete strength effect on damage index due to ultimate deformation and
hysteretic energy dissipation. ...................................................................................................92
Figure 4.1 (a) Elevation of the pier test units; (b) Cross section of DPT unit with
reinforcement details. ...............................................................................................................97
Figure 4.2 Reinforcement at BPT column base with foam at the base of the boundary
elements and secondary longitudinal steel in the wall. ........................................................98
Figure 4.3 Concrete material strength development curves.........................................................99
Figure 4.4 BPT Test setup at the NEES Multiaxial subassemblage Testing (MAST)
laboratory................................................................................................................................. 101
Figure 4.5 Loading components and moment diagram............................................................. 102
Figure 4.6 (a) Curvature and shear deformation instrumentation, North-East elevation; (b)
Cross section with instrumentation on each side. ............................................................. 104
Figure 4.7 (a) DPT diagonal loading protocol and (b) BPT biaxial loading protocol. .......... 106
Figure 4.8 DPT web crushing failure at

=4 x 2 loading toward A. .................................... 110

Figure 4.9 BPT web crushing failure at

=6 loading toward B. ............................................ 111

Figure 4.10 Hysteretic loops of DPT: (a) longitudinal axis and (b) transverse axis. ............. 112
Figure 4.11 Hysteretic loops of BPT: (a) longitudinal axis and (b) transverse axis. .............. 113
Figure 4.12 Hysteretic loops of BPT at
=4: (a) longitudinal axis and (b) transverse axis.
................................................................................................................................................... 114
Figure 4.13 Curvature profiles of BPT and DPT in different directions. ............................... 119
Figure 4.14 DPT in-plane minimum principal strain, E2 at

=4x1...................................... 120

Figure 4.15 BPT in-plane minimum principal strain, E2 at

=4xA. .................................... 121

Figure 4.16 Comparison of the BPT force-displacement envelopes in different directions.124
Figure 4.17 DPT hysteretic loop compared to BPT envelopes in diagonal and sweeping
diagonal direction. .................................................................................................................. 124
xii

Figure 4.18 Hysteretic loops of DPT in (a) flexure and (b) shear. ........................................... 125
Figure 4.19 Hysteretic loops of BPT in (a) flexure and (b) shear. ........................................... 126
Figure 4.20 Multidirectional loading effects: (a) Shear unloading stiffness degradation of
DPT in diagonal direction, and (b) Shear secant stiffness degradation of BPT in
transverse direction. ............................................................................................................... 127
Figure 4.21 Hysteretic loops of Unit 3A (a) Global behavior (b) Shear behavior (Hines 2004).
................................................................................................................................................... 128
Figure 4.22 Comparison of shear dissipating energy of pier tests and single wall test. ........ 129
Figure 5.1 Overview of 3D ABAQUS model: (a) Model geometry; (b) Reinforcement part;
(c) Mesh of the concrete part ............................................................................................... 137
Figure 5.2 M05M global force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 149
Figure 5.3 M10M global force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 149
Figure 5.4 M15M global force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 150
Figure 5.5 M20M global force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 150
Figure 5.6 M05M flexural force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 151
Figure 5.7 M05M shear force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 151
Figure 5.8 M20M flexural force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 152
Figure 5.9 M20M shear force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 152
Figure 5.10 M05M contour plot: Compressive damage variable (DAMAGEC) ................... 153
Figure 5.11 M20M contour plot: Compressive damage variable (DAMAGEC) ................... 153
Figure 5.12 M05M curvature profile comparison between experiment and ABAQUS. ...... 154
Figure 5.13 M20M curvature profile comparison between experiment and ABAQUS. ...... 154
xiii

Figure 5.14 M05M contour plot: Compressive equivalent plastic strain (PEEQ). ................ 155
Figure 5.15 M20M contour plot: Compressive equivalent plastic strain (PEEQ). ................ 155
Figure 5.16 M05M: The min. principal stress-log. strain curve of a crushed element. ......... 156
Figure 5.17 M20M: The min. principal stress-log. strain curve of a crushed element. ......... 156
Figure 5.18 VecTor2 finite element mesh for single walls ........................................................ 157
Figure 5.19 Finite element model with load cases for single walls in VecTor2. .................... 160
Figure 5.20 M05M global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 162
Figure 5.21 M10M global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 162
Figure 5.22 M15M global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 163
Figure 5.23 M20M global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 163
Figure 5.24 M05M deformed model at web crushing failure. .................................................. 164
Figure 5.25 M15M deformed model at web crushing failure. .................................................. 164
Figure 5.26 M05M direction of principal compressive strain (ε2) at failure. .......................... 165
Figure 5.27 M15M direction of principal compressive strain (ε2) at failure. .......................... 165
Figure 5.28 M05M: The principal compressive stress-strain curve of a crushed element.... 166
Figure 5.29 M20M: The principal compressive stress-strain curve of a crushed element.... 166
Figure 5.30 M05C global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 170
Figure 5.31 M10C global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 170
Figure 5.32 M15C global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 171
Figure 5.33 M20C global force-displacement behavior comparison between experiment and
VecTor2. .................................................................................................................................. 171
xiv

Figure 5.34 M10C deformed model at web crushing failure. ................................................... 172
Figure 5.35 M20C model with magnified deformation at

=4 x 2. ...................................... 172

Figure 5.36 M05C: The principal compressive stress-strain curve of a crushed element..... 173
Figure 5.37 M20C: The principal compressive stress-strain curve of a critical element. ...... 173
Figure 5.38 M20C flexural force-displacement curves from experiment and VecTor2. ...... 176
Figure 5.39 M20C shear force-displacement curves from experiment and VecTor2. .......... 176
Figure 5.40 M20C VecTor2 analysis: comparison of the shear deformation between the
bottom and top panel. ........................................................................................................... 177
Figure 5.41 M20C global force-displacement behavior comparison under cyclic and
monotonic loading with VecTor2........................................................................................ 177
Figure 6.1 M05C hysteretic behavior with failure predictions. ................................................. 184
Figure 6.2 M10C hysteretic behavior with failure predictions. ................................................. 185
Figure 6.3 M15C hysteretic behavior with failure predictions. ................................................. 185
Figure 6.4 M20C hysteretic behavior with failure predictions. ................................................. 186
Figure 6.5 Concrete compression softening parameter at web crushing failure for single walls
under cyclic loading: (a) Original Hines-Seible model; (b) Modified Hines-Seible model.
................................................................................................................................................... 189
Figure 7.1 The analytical force-displacement responses in principal and diagonal directions.
................................................................................................................................................... 192
Figure 7.2 Nonlinear finite element model of diagonal pier test (DPT) unit: (a) Model
overview; (b) Reinforcement; (c) Mesh of concrete parts ................................................ 194
Figure 7.3 DPT: global force-displacement behavior comparison between experiment and
ABAQUS. ................................................................................................................................ 197
Figure 7.4 DPT Unit contour plot: Compressive equivalent plastic strain (PEEQ). ............ 197
Figure 7.5 DPT Unit contour plot: Tensile equivalent plastic strain (PEEQT). ................... 198
Figure 7.6 DPT Unit contour plot: shear stress component in horizontal plane (S, S13). .. 198
Figure 7.7 DPT Unit contour plot: shear stress component in horizontal plane (S, S23). .. 199
xv

Figure 7.8 Equivalent single wall concept for the capcity analysis of hollow piers. .............. 200
Figure 7.9 DPT Unit hysteretic behavior with failure predictions. .......................................... 203
Figure 7.10 BPT Unit hysteretic behavior with failure predictions. ........................................ 203

xvi

Chapter 1.
1.1

Introduction

Motivation
The construction of the three new toll bridges in California, the Benicia-Martinez

Bridge, the Carquinez Bridge and the San Francisco-Oakland Bay Bridge (SFOBB), has
highlighted the relevance of hollow rectangular bridge piers with highly-reinforced boundary
elements and connecting walls for the piers of high-profile bridges. The proof-of-concept
test of the 1/4-scale SFOBB piers showed that the pier design can exhibit excellent ductile
flexural behavior when subjected to severe loading protocols [1][2]. However, the
conservative design results in a massive cross section which can make the construction even
more expensive. The modification of the design into a more slender cross section by making
use of high-strength-concrete and new concepts for shear capacity was recently hypothesized
by Hines [3] at the conclusion of the SFOBB proof-of-concept testing. Figure 1.1 shows the
cross sections of SFOBB pier test unit with the current design compared to a potentially
more slender design as hypothesized by Hines. However, the noted hypothesis lacked
experimental verification on the seismic performance of such a slender pier. Especially
needing validation is the fact that the potentially slender design hypothesis is rooted in the
assumption that increased shear capacity can be obtained from the walls by making use of
high-strength-concrete without compromising ductility capacity.
The in-plane behavior of the hollow bridge piers can be well characterized through
the study of individual structural walls. The behavior and failure mode varies with aspect
ratio. section geometry and reinforcement configuration. For slender shear walls of
multistory buildings, the high flexural demand tends to result in flexural failures in the form
of the fracture of the tensile reinforcement or crushing of the concrete in the compression
1

zone. For low-rise, or squat, shear walls the possible failure modes include diagonal tension
and compression failures, instability, and sliding shear failures. Web crushing, or diagonal
compression failure, manifests itself as compression failure of concrete struts formed by
diagonal tension cracking in the wall web. The failure is brittle in nature and therefore, it is
mandated to be suppressed during design.

24”

84”

32”

9”

84”

6”
R8.5”

R11.5”
54”

54”

Figure 1.1 SFOBB pier test unit: Present design (Left), Potential design (Right).
Among the structural walls with different cross sections, flanged and barbell shaped
walls have been found to be more susceptible to web crushing [4][5]. It has been realized the
shear distortion behavior needs to be well investigated in order to study the web crushing
behavior of the structural walls. Inelastic shear behavior, exhibited in common structural
walls, is characterized by rapid strength and stiffness degradation and hence limited energydissipating capacity. In earthquake resistant design, such behavior is undesirable and the
2

corresponding failure modes should be avoided. However, in structural walls with wellconfined boundary elements, web crushing failures can take place after adequate ductile
response is attained. The key to obtain higher ductility level is that the load-transfer
mechanism formed within the structural component should be maintained, that is, the
diagonal truss effectively carrying the shear force should be efficient in transferring the load
from the tension boundary element to the compression boundary element. It can thus by
hypothesized that high-strength-concrete (HSC) can forestall the web crushing failure and
increase the displacement capacity of the structural walls since HSC struts should be able to
keep the load-transfer mechanism at higher demands.
The response of reinforced concrete elements limited by shear failure but displaying
stable inelastic response with adequate energy dissipation characteristics can give rise to a
new genre of ductile failure mechanism that may be termed as “ductile shear failure” with a
two-fold meaning. First, like ductile flexural failures, a ductile shear failure conforms to the
design philosophy appropriate for seismic regions that require a stable hysteretic response
able to dissipate energy up to large levels of inelastic deformation before failure.
Experimental studies since the mid-seventies of last century [5][6] have demonstrated that
structural walls with well-confined boundary elements or flanged sections could exhibit
adequate ductile behavior before web crushing failure in the plastic hinge region at
displacement ductility levels of four or greater [7]. By establishing reasonable performance
limits, ductile shear failures may control the strength and ductility of the structural
components for design purposes. Second, failure to consider the shear stiffness degradation
of the inelastic flexure-shear region under load reversals could result in an unsafe design
according to capacity design procedures. The possibility that flexure controlled structures
failing in shear thus arises. Unfortunately, the shear strength and stiffness degradation
3

mechanism of structural walls under coupled inelastic flexure-shear demands is not well
understood.
High-strength-concrete (HSC) offers the potential in optimizing structural design
and reducing material cost. The potential benefits of HSC bring about new possibilities for
the hypothesized more slender designs. First, HSC can provide enough capacity for the
compression zone of the boundary element under moment demand. The confinement
provided by the spiral will further enhance the strength and ductility of the confined
concrete. It should be noted that the fracture failure of the longitudinal reinforcement in the
tension boundary element is not considered. Otherwise, the design needs to be modified
with higher longitudinal reinforcement ratio. This can be easily checked with sectional
moment-curvature analysis. Thus, a more slender design with comparable moment capacity
can be obtained by using the same global dimensions of the cross sections and the capacity
of the compression zone. Second, the disadvantage of the potential design is that the shear
capacity is decreased due to the more slender connecting walls. Thus, possible shear failure
modes could happen. Even though the diagonal tension capacity can be satisfied with the
configuration of the wall transverse steel, the possibilities of the web crushing could still
exists.
The seismic performance of structural walls or wall assemblies is highly dependent
on the ductile behavior and energy dissipating capacity of the plastic hinge regions.
According to capacity design principles, inelastic shear damage mechanisms should not be
relied upon due to their rapid shear stiffness degradation and their inferior energy dissipating
capacities. However, recent research [8] has demonstrated that pier walls under shear
demands considerably above current design limits can exhibit stable ductile behavior before
experiencing web crushing failures, and that their inelastic deformation capacity can be
4

further improved by using high-strength-concrete (HSC). Such facts thus lead to the
suggestion that inelastic flexure-shear behavior with subsequent inelastic shear failure can
satisfy seismic performance requirements on bridge piers. It also has to be recognized that
the use of HSC entails careful damage assessment to meet serviceability objectives and
explicit criteria on performance limits to ensure meeting safety.
The vulnerability of structures subjected to earthquake motions along random
directions makes the performance evaluation of wall assemblies under multi-directional
loading necessary. It is simple to understand that the damage on the web concrete will
accumulate with loading in different directions. Some research has been conducted to study
the behavior of the three-dimensional shear walls in T, L or I shape [9][10]. However, the
demand and capacity of the walls in non-principal directions has not been fully investigated.
The dispersion of the tensile forces from the boundary elements or flanges makes it difficult
to decide the effective dimension of the web, which plays an important role in the loadtransfer mechanism. Characterization of the three-dimensional web crushing behavior is
therefore essential in order to evaluate the web crushing capacity of piers along nonprincipal directions of the cross section.

1.2

High-strength-concrete in seismic regions
Modern concrete manufacturing technology has made the industrial production of

high-strength-concrete (HSC) up to 138 MPa (20 ksi) available with the evolution of the
silicon fume and chemicals of retarders, superplasticizers, etc. There is no doubt that the use
of HSC will continue to increase. However, the lack of the design provisions, especially for
the use of HSC in seismic design prompts the need for research in this area.

5

On one hand, the application of HSC could make the designed structural
components be more slender by reducing the cross-section dimensions and by consequence
reduce seismic mass and reduce the size and cost of substructure units. The compactness
and low creep and shrinkage of HSC would improve the durability of the structures under
various weather conditions. On the other hand, HSC could jeopardize the ductility level of
the structures in seismic regions due to the rapid strength degradation within a small strain
range. Therefore, careful evaluation is needed in order to satisfy functional and safety seismic
design criteria. Moreover, premature cracking of HSC elements needs appropriate evaluation
to satisfy performance-based requirements.

1.3

In-plane Web Crushing Behavior
A membrane panel under uniform shear stresses along all sides will exhibit parallel

shear cracking in the diagonal direction. With the occurrence of axial compression stresses,
the cracking angle tends to be less than 45º from the vertical. Web crushing will happen
when the compression capacity of the strut is reached. This mechanism, consisting of web
crushing failure with parallel shear cracking can be referred to as an “elastic web crushing
failure.” Structural walls that fail in such kind of mechanism have very limited deformation
capacity. Thus, current shear design provisions restrict elastic web crushing failures by
providing specifications on the configuration of the transverse steel and thus limit the shear
strength of the walls by a diagonal tension failure, which can be seen reviewing the ACI-318
shear design provisions.
Recent research has revealed that structural walls with flanges or boundary elements
develop a fanning crack pattern in the plastic hinge region under moderate or large inelastic
deformations. It has also been noticed that with the increase of the deformation demand, the

6

diagonal cracks realign in the region of the web next to the compression boundary element.
When walls experience large rotations the lower struts become less effective in their load
transfer efficiency. Consequently, the realignment of crack results in the formation of a small
region with high compressive stresses where web crushing tends to happen. In this study,
the web crushing failure mechanism with realigned fanning crack pattern is referred to as an
“inelastic web crushing failure”.
The just described elastic and inelastic shear cracking patterns can observed in Figure
1.2, which shows a cantilever structural wall test unit that failed by web crushing. The
realigned fanning cracks in the plastic hinge region are highlighted, while a strut formed by
elastic shear cracking in the web above the plastic hinge region is also highlighted. It can be
seen that the flexural cracking on the tension boundary element appears to be flat. In the
inelastic shear cracking region of the web, the change of the crack angle can be observed by
comparing the inclination of the bottom and top edges. It is reasonable to assume that the
smaller the crack angle from the vertical is, the more efficient the strut work to transfer the
load from the tension boundary element to the compression boundary element. However,
with inelastic deformation and the development of the plastic hinge length the cracking close
to the base realigns due to the evolution of the flexural cracking into the web. Thus, the
inelastic shear cracking develops due to the combination of the flexure and shear effect. The
term of “flexure-shear effect” is thus used in this study to refer to such behavior. This effect
makes the struts near the wall base less effective in load transfer since the crack angle (about
the vertical) is increasing. Hence, the top-most strut in the inelastic shear region next to the
elastic shear cracking region is the most efficient but critical strut for limiting web crushing,
as is shown in Figure 1.2 (c).

7

(a)

V P

P
P

V
V

P
P

(b)

(c)

V

V

Elastic
Shear

F1

s
s

Av fy

ΔT

s

F1

s
s

Inelastic
Shear

Av fy
f s

f s


L pr

L pr

s

ΔT

F2

dθ

d

R

R
V
T

T

jd

C

C

F2

V

Figure 1.2 Flexure-shear and elastic shear mechanisms: (a) In test wall, (b) Sketch of the
regions, (c) Truss models.
However, it has been observed that actual critical region limiting web crushing under
cyclic loading may be different. For cyclic loading the failure tends to happen at the height
level of the bottom strut in the fanning pattern. It is hypothesized that this is due to the
criss-crossed cracking induced by the cyclic loading. For HSC structural walls, the web
experiences more densely-distributed cracking due to the brittle nature of concrete. Thus,
the concept of well-defined critical struts is more of an ideal assumption than an
experimental observation.

1.4

Diagonal Web Crushing Behavior
The web crushing capacity of wall assemblies is complicated due to the spatial

distribution of boundary elements and connecting walls. In addition to the response along
the principal directions (along the sides), the web crushing capacity along non-principal, or
diagonal, directions is of interest in order to characterize the three-dimensional failure
8

behavior. Under loading in the diagonal direction, extreme and intermediate boundary
elements with respect to a non-principal axis of bending will be in tension while as little as
one element may be in compression. The existence of intermediate boundary elements with
respect to the line of bending will disturb and separate the wall shear cracking. In this study,
the evaluation of web crushing for wall assemblies was evaluated within the context of a
square hollow pier with corner boundary elements. It has been observed the response of
such pier units under diagonal loading will cause the crack angle of the tension wall to have a
bigger cracking angle from the vertical; while in the compression side the cracking angle is
smaller. The coexistence of large tensile stresses due to flexure with the shear demands is the
reason for the cracking being relatively flat. The determination of the distribution of the
shear stress on the walls between the tension boundary elements (back walls) and the walls
between intermediate and compression boundary elements (front walls) is the main issue. It
is reasonable to assume that the front walls carry more shear stress than the back walls.
Nonetheless, further investigation is needed in order to analyze the diagonal, or nonprincipal, web crushing capacity of wall assemblies.

1.5

In-plane Cyclic Loading Effect
It seems straightforward to assume that web concrete in structural walls will degrade

more under cyclic loading than monotonic loading. Some research has been conducted to
study the damage of concrete under complicated stress states with loading history
considered. However, further investigation is needed on the degradation of HSC under
cyclic loading.

9

1.6

Multi-directional Loading Effect
The vulnerability of structures subjected to earthquake motions along random

directions makes the performance evaluation of wall assemblies under multi-directional
loading necessary. Although noteworthy research has been done on the behavior of
rectangular hollow bridge piers under biaxial loading, most of it was focused on assessing
biaxial flexural behavior and slenderness limits [11]. Mo et al. [12] studied the seismic
performance of hollow bridge columns under uni-directional cyclic loading. Takizawa et al.
[13] and Bousias et al. [14] studied the biaxial flexural behavior of pier columns under
various loading patterns with the aim of establishing the interaction, or coupling, effects
between the two perpendicular directions. Hines et al. [1][2] conducted two proof-ofconcept tests on a typical hollow rectangular skyway pier of the new San Francisco-Oakland
Bay Bridge (SFOBB), which focused on evaluating a pier design with highly-reinforced
boundary elements and connecting walls subjected to uni-directional and bi-directional cyclic
loading. All of the noted efforts were related to the flexural behavior of the hollow
rectangular piers, which can be seen as a closed assembly of walls with corner boundary
elements, and none of them considered shear demands and shear failure modes under multidirectional loading. A related investigation related to shear behavior is the work by Wong et
al. [15] who studied the ductility level and different failure modes of circular columns with an
aspect ratio of 2. Test results revealed that biaxial displacement patterns led to severe
strength and stiffness degradation compared to uniaxial displacement and that shear failures
with moderate ductility could be achieved. Thus, even though the mechanisms behind
inelastic plane shear web crushing have been reported for some time [5][6][7][16][17], as is
described in Section 3.4, the three-dimensional shear resistance and failure mechanisms of
wall assemblies and hollow rectangular bridge piers is still not fully investigated.
10

1.7

Objectives
The objective of the study on the in-plane web crushing capacity of the structural

walls is to examine the use of high-strength-concrete (HSC) to obtain ductile shear failure
behavior and evaluate the hypothesis that web crushing capacity is related to concrete
compressive strength

instead of the √ . The approach is non-intuitive as web crushing

is a brittle failure and it is well known that concrete becomes more brittle as its compressive
strength increases. The effect of cyclic loading on concrete damage, structural
strength/stiffness degradation, and web crushing capacity is to be evaluated. The appropriate
performance evaluation on HSC structural walls exhibiting ductile shear failure is expected
to contribute to the groundwork of the next stage in earthquake resistant design of thinwebbed elements and systems. Moreover, the study of web crushing as a fundamental shear
failure mechanism can have potential impact on present shear design provisions.
The objective of the study on wall assemblies is to examine the application of HSC
to achieve ductile shear failure behavior, evaluate the diagonal web crushing capacity and the
concrete degradation under multi-directional effect in order to fully evaluate seismic
performance. The behavior of wall assemblies is to be studied within the context of hollow
square bridge piers, which are composed of a closed arrangement of walls with corner
boundary elements. The study is aimed at corroborating that HSC can increase the web
crushing capacity of wall assemblies, to determine the three-dimensional shear demands on
wall webs, and to assess the effect of multi-directional loading on the degradation of the
shear web crushing resisting mechanism. The diagonal web crushing capacity will be
evaluated and analyzed, which can be further applied on the analysis of the threedimensional walls with cross sections in other shapes.
11

1.8
1.8.1

Approach
Experimental studies on single structural walls
The in-plane seismic performance of single HSC structural walls featuring highly-

reinforced boundary elements and a thin constrained web was evaluated through
experiments on eight 1/5-scale single walls designed with concrete compressive strengths of
34, 69, 103, and 138 MPa (5, 10, 15 and 20 ksi) and tested under cyclic and monotonic
loading protocols. The objective of this study was to evaluate the effect of concrete strength
on the web crushing performance limits of HSC structural walls. The web crushing behavior
of the structural walls with the same design concrete compressive strength but under
different loading protocols: monotonic and cyclic loading, was compared to identify the
seismic effect on the ductility level of the walls. The experimental results demonstrated that
high-strength-concrete can effectively delay web-crushing failures and increase the
displacement ductility levels of the structural walls. However, web crushing capacity was
significantly curtailed due to the rapid shear strength and stiffness degradation under cyclic
loading. Nonetheless, ductile shear failures were still attained for HSC structural walls up to
intermediate displacement ductility levels.
1.8.2

Experimental studies on wall assemblies
To address the knowledge gap on the shear response of wall assemblies and evaluate

the potential of HSC to allow for ductile shear failures as an acceptable inelastic failure
mechanism on such systems, two 1/4-scale hollow square bridge piers subjected to multidirectional cyclic loading with design concrete compressive strengths of 34 and 138 MPa (5
and 20 ksi) were tested. The three-dimensional inelastic web crushing behavior was evaluated
from observations and measured hysteretic force-displacement behavior. Loading path
12

effects on web crushing capacity and the degradation of shear stiffness and energy
dissipating capacity were quantified. Test results support that HSC can delay web crushing
shear failures, thus allowing for the dependable inelastic response of hollow bridge piers.
However, the mixed flexure-shear cracking under multi-directional loading causes rapid
shear stiffness degradation, an effect that is amplified for HSC due to its lower fracture
toughness, which curtails the web crushing resistance capacity of the wall assembly.
1.8.3

Nonlinear finite element modeling
Nonlinear finite element modeling (NLFEM) was conducted to assist in the

understanding the web crushing behavior of the structural walls and bridge piers. This study
gave an appropriate evaluation on the capacity and limits of present finite element tools in
evaluating the flexure and shear force-displacement behavior and capacity of reinforced
concrete walls. The study provided quantitative data on the stress and strain state of the web
crushing region and assisted the development of simplified analytical models.
1.8.4

Simplified analytical modeling
A simplified analytical method based on test observations and a truss analogy

approach was developed to study the web crushing capacity of the HSC structural walls and
hollow bridge piers. The inelastic web crushing model by Hines and Seible formed the basis
for this task. However, the model was modified based on the learned response of HSC walls.
The test results from prior tasks in this study were be used to calibrate the model.

13

1.9

Scope
This study focuses on the web crushing behavior and assessment of HSC structural

walls and the potential to define ductile shear failure as a viable failure mechanism for
seismic design. As such, the work aims to demonstrate that the traditional opinion that shear
failure mechanisms are inferior in seismic performance is not always true and that a
dependable ductile failure mechanism ultimately limited by shear (web crushing) is possible.
To evaluate and analyze the web crushing capacity of the in-plane structural walls and wall
assemblies, the scope of this study was divided into four tasks: First, the in-plane web
crushing behavior of the HSC structural walls was investigated through experiment to
corroborate that ductile web crushing behavior can be achieved. Second, to evaluate the
seismic performance of the HSC wall assemblies, the diagonal web crushing behavior and
accumulation of concrete damage under multi-directional loading effect was investigated
through hollow pier tests. Third, nonlinear finite element modeling was conducted to assist
in the understanding the test observations and results both phenomenologically and
quantitatively. Finally, a simplified analytical method is proposed to the engineering field for
design purposes.

1.10 Organization
The dissertation has been organized into eight chapters. Chapter 1 provides the
motivation of this study, and then highlights the key research aspects of high-strengthconcrete in seismic regions, in-plane web crushing behavior, diagonal web crushing behavior,
in-plane cyclic loading effects and multi-directional loading effects. The objective of the
study is addressed followed by the research approach, with the combination of the

14

experimental methods on structural walls and hollow bridge piers, nonlinear finite element
modeling and the simplified analytical method for design.
Chapter 2 reviews general shear strength models and the shear design provisions of
US codes. The web crushing capacity models of Oesterle et al., Paulay and Priestley and
Hines et al. are reviewed and discussed. The theories and implementation of the finite
element approaches used in the work are presented. Finally, the research hypothesis of this
study is introduced.
Chapter 3 reports on the experimental studies of web crushing performance of inplane loaded HSC structural walls. The experimental investigation on single wall test units is
described, including details of materials, instrumentation, and loading protocols.
Experimental observations and results are described followed by discussion and conclusions.
Chapter 4 reports on the experimental studies of HSC hollow bridge piers. The wallassembly test units, materials, test setup, instrumentation and loading protocol are described.
Experimental results are provided, followed by discussion and conclusions.
Chapter 5 presents the study on nonlinear finite element modeling of HSC structural
walls. A general commercial nonlinear finite element program (ABAQUS) and a research
nonlinear finite element analysis for reinforced concrete membrane structures (VecTor2)
were used to model the web crushing behavior of the structural walls under monotonic
loading and cyclic loading. The analyses are described in the sequence of geometrical
modeling, definition of material properties, solution controls and results. Discussions and
conclusions on the results and the applicability and limitations of the noted numerical tools
to capture the inelastic response of reinforced concrete walls are provided.
Chapter 6 reports on a simplified analytical method to assess the web crushing
capacity of structural walls based on the inelastic web crushing model by Hines and Seible. A
15

modification to the approach by Hines and Seible is proposed to calculate the softening of
web concrete. The modified model is assessed by comparing its prediction to the
experimental response of the HSC structural walls tested in this study. A discussion and
conclusions are provided at the end.
Chapter 7 describes a simplified analytical method to assess the diagonal web
crushing capacity of HSC hollow bridge piers. An equivalent single wall in the diagonal
direction was exacted and analyzed. The modified Hines model presented in Chapter 6 was
implemented for the three-dimensional web crushing capacity analysis.
Finally, Chapter 8 provides a summary of the research findings, the original
contributions and research impact of the study, and the future research needs.

16

Chapter 2.
2.1

Literature Review and State-of-the-art

Summary
In this chapter, an approach for estimating the force-displacement behavior of

reinforced concrete structural walls is described first. Then, a historical review of shear
strength models is provided. The shear design provisions of US codes are reviewed with the
maximum allowable design shear stress being emphasized. The web crushing capacity
analytical methods of Oesterle et al., Paulay and Priestley and Hines and Seible are reviewed
and discussed. With respect to the nonlinear finite element modeling (NLFEM) effort, the
Modified Compression Field Theory (MCFT) and the implementation of NLFEM in the
program VecTor2; and the continuum damage plasticity theory for concrete implemented in
the program ABAQUS are described. Finally, the research hypothesis of this study is
presented.

2.2 Force-Displacement Behavior of Structural Walls
The deflection of a cantilever member can be calculated by integrating the curvatures
along the height of the member [18] by:
(2-1)

∫
where

is the height of the cantilever member; and

segment of length

is the distance of to an infinitesimal

from the free end. The tip deflection of an element can thus be

calculated based on the moment-curvature relationship of its cross-section. Equation (2-1) is
usually simplified based on the idealization of a linear curvature distribution in the elastic and
inelastic regions. For this simplification, the inelastic, or plastic, curvatures are taken to be
linearly distributed over an equivalent plastic hinge length,
17

. The estimate of plastic

rotations and plastic deformations of a member is highly relied upon for the definition of the
plastic hinge length

.

In this study, the approach proposed by Hines et al. [19] was used to calculate the
plastic hinge length and the force-displacement behavior of structural walls. Three distinct
phenomena affecting the spread of plasticity in reinforced concrete members are considered:
moment gradient, tension shift, and strain penetration. Assuming that plastic rotation occurs
about the column base,

is evaluated as
(2-2)

where

is the length of the plastic hinge region (or extent of plasticity); and

length of the strain penetration (or bond slip region).

√

(

is the

is calculated as:

(2-3)

)

The displacement of the wall up to the first yield is calculated as:
(

).

/

(2-4)

After first yield, the displacement is calculated as:
.
where

is equal to

,

/
and

.

/

(2-5)

are the diameter and yield strength of the

longitudinal steel. The ratio of the flexural to shear deformations was taken as 0.24 based on
the single wall tests by Hines et al. [17] and this study described in Chapter 3 [8]. The values

18

for

,

and

are the corresponding values of displacement, moment and curvature at

first yield, defined as the first occurrence of yield in the extreme steel bar in tension.

2.3 Historical Overview of Shear Strength Models
The equivalent truss model was been widely used since the 1900‟s to understand the
shear behavior of reinforced concrete beams with transverse reinforcement. The 45º truss
model was developed by Ritter in 1899 [20] and refined by Mörsch in the 1920‟s [21][22]. In
1907, Withey introduced the 45º truss model to the US and reported that compared to
experimental results, the model is conservative [23]. In the late 1950‟s, the 45º truss model
was also applied to prestressed concrete structures.
In 1962, the ACI-ASCE Joint Committee 326 on shear and diagonal tension
recommended an empirical equation for the shear strength contributed by concrete [24],
which is still used in ACI-318-08 [25].
(

)

√

(2-6)

In 1974, Collins and Mitchell developed the compression field theory (CFT) where
equilibrium equations, compatibility for averaged concrete and reinforcement strains, and a
constitutive relationship for cracked concrete and reinforcement are considered [26][27].
Since CFT is still a truss model, no shear stress transfer at the crack interface is considered
and the concrete tensile constitutive relationship is neglected.
In 1986, the modified version of CFT, the modified compression field theory
(MCFT) is proposed by Vecchio and Collins [28]. In this refined model, shear stresses at
cracks are estimated and concrete tension stiffening effects are included. After parametric
calibration with a series of membrane tests, MCFT represents a robust model to predict the
shear force-displacement behavior of reinforced concrete members.
19

Similar to the concept of MCFT, in the 1990‟s, Hsu et al. developed the rotatingangle softened-truss model (RA-STM) [29][30] and the fixed-angle softened truss model
(FA-STM) [31]. The development of these approaches deviates from the MCFT in that no
assumption is made with regards to the principal stress and strain directions of stress in the
concrete. However, the models are similar in considering a bi-axial state of plane stress on a
reinforced concrete element and enforcing equilibrium and compatibility requirements for
the cracked reinforced concrete. In this study, the general framework of the MCFT is used.

2.4 US Code Provisions for Shear
2.4.1

ACI-318-08 Shear Provisions
In ACI-318-08 [25], except for prestressed members and members designed based on

strut-and-tie models, the nominal shear strength,
contributions from concrete

and shear reinforcement

, is computed as the sum of the
, that is,
(2-7)

For normal weight concrete walls subject to axial compression
√

(2-8)

unless a more detailed calculation is made. Because of the lack of test data and practical
experience on high-strength-concrete (HSC), mainly with compressive strength over 10,000
psi, a maximum value of 100 psi has been placed on √

since the 1989 edition of the code.

The shear strength provided by steel is calculated based on the 45º truss model and is
expressed by:
(2-9)

20

Even though there is no explicit limit of √

on

as in ACI-318-02, the same limit on

is maintained, that is,
(2-10)

√

However, it is admitted in the commentary of the corresponding provision that shear
stresses in excess of
2.4.2

√

can be obtained.

AASHTO Standard Specifications for Highway Bridges
The AASHTO Standard specifications [32] follow essentially the same shear design

provisions of ACI-318 on reinforced concrete members. The maximum allowable limits on
shear stresses from ACI-318 are maintained.
2.4.3

AASHTO LRFD Bridge Design Specifications
Since 1994, the Sectional Design Model based on the Modified Compression Field

Theory (MCFT) has been introduced into US bridge design practice [33]. When using the
model, the axial strain in the member at mid-height is evaluated under the effect of axial
load, moment, prestressing, and shear. The strain and the shear design stress level are used
to select values of the coefficients

and

from tables. These values, obtained by

calculations from the CSA method [34], control the contributions of concrete and steel to
the shear resistance. The sectional design model is more complicated than the shear design
provisions in AASHTO‟s Standard Specifications, which often requires an iterative solution.
Equation (2-7) is still valid for non-prestressed members. The concrete contribution
of shear forces is calculated as:
√

21

(2-11)

The contribution from the shear reinforcement is calculated based on the parallel
chord truss model with the cracking angle

being a variable,
(2-12)

With the shear design stress ratio
of the cross section, the values of

and

The nominal shear resistance

⁄

and the longitudinal strain

can be obtained from a table.
has an upper limit, defined by
.

2.4.4

at mid-height

(2-13)

Discussion
Historically, the 45º truss model for the analysis of cracked reinforced concrete

members was been widely used since the early 1900‟s. The corresponding ACI shear strength
provision of Equation (2-6) came into shape in 1960‟s, and shortly after the 45º truss model
was re-examined since it is overly conservative. It is thought-provoking to notice that the
same model is still in use in the up-to-date design provisions in the twenty-first century.
There are two types of shear cracking that may take place in reinforced concrete
beams, which are described in ACI-318, and can be referred to for the cracking response of
reinforced concrete walls. One is web-shear cracking and the other is flexure-shear cracking.
Web-shear cracking begins from an interior point in the web when the principal tensile
stresses exceed the tensile strength of the concrete. Flexure-shear cracking is initiated by
flexural cracking. When flexural cracking occurs, the shear stresses in the concrete above the
crack are increased. Flexure-shear crack develops when the combined shear and tensile
stresses exceed the tensile strength of the concrete. For the structural walls in this study,
web-shear cracking happened at half the load required to produce flexural yielding in the

22

extreme tensile reinforcement. With increasing displacement demand flexure-shear cracks
develop and crack-realignment was observed.
It can be seen that ACI-318 shear provisions are oriented and aimed at diagonal
tensile cracking and failure. The possibility of the web crushing failure is not considered
when diagonal tension failure is well resisted in a conservative design. Since the code does
not reflect the direct dependence of web crushing capacity on

, but rather limits shear

demand based on √ , the application of the shear provisions on evaluating the web
crushing capacity of the walls is problematic. Moreover, the lack of experience and
experimental data on the application of HSC hinders the possible efficient design of
reinforced concrete elements limited by web crushing failure.
In AASHTO LRFD, a maximum allowable shear stress of

is prescribed,

which is very different from the recommendations in ACI-318-08 and the AASHTO
Standard Specifications. This difference is more significant with the increase of concrete
compressive strength. As is the case in the ACI-318 recommendations, the upper limit is
intended to avoid web crushing failure such that diagonal tension failure occurs first. It is
also specified in the AASHTO LRFD Specifications that for concrete strengths over 10.0
ksi, physical tests or specific articles are needed to establish the relationships between the
concrete strength and other properties.
Figure 2.1 shows the analytical force-displacement responses [19] of the test units,
with varied nominal concrete strengths from 5 ksi to 20 ksi and the corresponding lines for
the ACI-318 shear stress limits. The description of the test unit is included in Section 3.2.
The values of the lines are calculated purely according to
√

√

and the limit of 100 psi on

is not applied. It may be deceiving to see that the ACI-318 limit gives a conservative

23

estimate to the ideal capacity of the walls. However, two aspects need to be pointed out.
First, that the force-displacement response is an analytical prediction of the inelastic flexural
response of the wall and does not consider shear failure. Second, the estimate will be
considerably in error if the limit of 100 psi on √

is applied. It should be further noted that

the force-displacement responses are essentially unchanged as the flexural response is
dictated by the longitudinal reinforcement in the boundary elements and not the concrete
compressive strength. This follows from the heavy reinforcement in the boundary elements
in the test units in this study. The heavily reinforced boundary elements are necessary to
develop the necessary shear stress in the wall web and properly anchor the diagonal shear
struts in the compression toe of the wall. The increase of the moment capacity with HSC is
thus not significant.

Displacement (in.)
0

1

2

3

4

5

6

1200

300
250
200

800
=20 ksi
=15 ksi
=10 ksi
=5 ksi

600
400
200

F-D Response

0
0

20

40

60

80

100

120

150
100

Force (kips)

Force [kN]

1000

50
0

140

Displacement [mm]
Figure 2.1 Analytical force-displacement responses of the single walls with ACI-318 shear
stress limits.
24

2.5 Web Crushing Capacity Models
2.5.1

Oesterle et al.
The research results by Oesterle et al. based on the experiments conducted by

Portland Cement Association (PCA) on structural walls [4][5] failing in web crushing are
summarized in this section.
Among the tested walls with flanged, barbell and rectangular cross sections, only
flanged and barbell walls were found to exhibit web crushing. From this observation, it can
be seen that the concentration of the resultant tension and compression forces is necessary
to form the load-transfer mechanism of the strut-and-tie model with the web demanded in
diagonal compression.
A fanning crack pattern in the plastic hinge region was observed on the walls. It was
also noticed that with the increase of the deformation demand, the diagonal cracks realigned
in the region of the web next to the compression boundary element. When specimen
experienced large rotations, the lower struts become less effective in load transfer. The
realignment of cracks resulted in the formation of a small region with high compressive
stresses where web crushing tended to happen.
An average effective compressive strut strength of 0.16~0.49

was obtained based

on the PCA wall tests. This value is deemed to be too low since a truss analogy with parallel
cracking and no realigned cracking was considered in its development. In view of this, the
calculation of the concrete softening parameter

proposed by Collins [27],
⁄

,

(2-14)

is not applicable for the inclined struts with realigned cracking geometry being considered.

25

The web crushing model by Oesterle et al. relate the shear strength

to the drift

ratio within the plastic hinge region, , by:
when

(2-15a)
(2-15b)

when
4

where

5

is the nominal shear stress at web crushing,

compressive strength,
⁄

, where

is the applied axial load, and

,

is the concrete

is the gross cross sectional area.

is the total deformation at the top of the plastic hinge region,

is the

length of the plastic hinge region.
Discussion
Figure 2.2 shows the analytical force-displacement responses [19] of the test units,
with varied nominal concrete compressive strengths from 5 ksi to 20 ksi and the
corresponding web crushing capacity curves as predicted by the model of Oesterle et al. The
description of the test unit is included in Section 3.2. It can be seen that the failure models
provide a more realistic dependence of web crushing capacity with inelastic deformation,
where as deformations increase, the web crushing capacity decreases. The shift of the
capacity curves with increasing concrete compressive strength is also obvious. It should be
noted, however, that the plotted curves have not implemented the limit proposed by
Oesterle et al., for the compressive struts of 0.16~0.49 , which would lead to dramatically
conservative capacity estimates.

26

Displacement (in.)
0

1

2

3

4

5

6

1200
=20 ksi

200

800
=15 ksi

600

=10 ksi

400

100

=5 ksi

50

200
F-D Response

0
0

20

40

60

80

100

150

Force (kips)

250

1000

Force [kN]

300

120

0

140

Displacement [mm]
Figure 2.2 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Oesterle et al.
2.5.2

Paulay and Priestley
To prevent premature diagonal web crushing failure from occurring before the

yielding of the shear reinforcement, a nominal shear stress is limited of

in the

plastic hinge regions of beams, columns, and walls was proposed by Paulay and Priestley [7].
It was realized from tests at PCA [5] and at the University of California, Berkeley [6], that
web crushing in the plastic hinge region could happen at displacement ductility levels of 4 or
more. When displacement ductility is equal to or less than 3, a shear stress over

could

be achieved. Paulay and Priestley [7], thus proposed a shear design equation that indicates
that web crushing capacity is directly proportional to concrete strength but degrades with
increased ductility:

27

.
where

/

is the displacement ductility ratio and

(2-16)

is the concrete compressive strength.

Discussion
Figure 2.3 shows the analytical force-displacement responses [19] of the single wall
test units from this research, with varied nominal concrete compressive strengths from 5 ksi
to 20 ksi and the corresponding web crushing capacity curves from the model of Paulay and
Priestley. The description of the test unit is included in Section 3.2. As discussed for the
Oesterle et al. model, the Paulay and Priestley model appropriately capture an increasing web
crushing shear strength with increasing concrete compressive strength and a degradation of
this capacity with increasing inelastic deformations. However, the plotted curves were also
removed from the proposed stress limit by Paulay and Priestly, which, if plotted would lead
to highly conservative estimates of shear strength. Compared to the mode by Oesterle et al,
the model by Paulay and Priestley is more conservative. The reason is that the Paulay and
Priestley model was intended mainly for design purposes while the model by Oesterle was
calibrated for assessment based on the test data from the PCA wall tests.

28

Displacement (in.)
0

1

2

3

4

5

6

1200

300
250
200

800

150

=20 ksi

600

=15 ksi

400

100

=10 ksi
=5 ksi

200

50
F-D Response

0
0

20

40

60

80

100

Force (kips)

Force [kN]

1000

120

0

140

Displacement [mm]
Figure 2.3 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Paulay et al.
2.5.3

Hines and Seible
The recent study by Hines and Seible [16] recognized a clear distinction in assessing

the capacities of elastic and inelastic web crushing based on the mechanisms noted in Figure
1.2. A flexure-shear model based on a truss analogy and the observed fanning crack pattern
inside the plastic hinge region was proposed, compared to an elastic and parallel shear
cracking pattern outside the hinge region. The demand of the critical strut in flexure-shear

(

) as well as the strut in elastic shear (

) was assessed based on equilibrium

consideration of the shown free body diagrams. The capacity of the critical flexure-shear
struts (

) was derived based on an appropriately formulated length of the plastic hinge

region (

), a concrete compression softening factor ( ) calculated based on the modified

29

compression field theory (MCFT) [28] and the estimated radius ( ) for the critical
compression strut fan.
The dimension of the strut in the elastic shear region is determined based on the
vertical width of the strut generated from flexure cracks at a spacing equaling the transverse
steel ( ). The dimension of the critical strut in the inelastic shear region is determined based
on the spacing of the transverse steel on the tension boundary element with its width at the
tension side defined through the calculation of the angle between the edges of the strut (

).

The derivation of the demand and the capacity of the strut in the elastic shear region are
shown in below.
The resultant tensile force on the strut (

) is calculated through by moment

equilibrium of the strut,
(
The demand on the strut,

)

(2-17)

can be calculated by considering force equilibrium

in the vertical direction,
(2-18)
The inelastic shear crack angle
of the plastic hinge region (

is calculated based on the estimate of the length

)
(2-19)

where

is calculated as [19]

30

(2-20)

√

To calculate the capacity of the strut
compression boundary element (

, the width of the strut next to the

) and the concrete softening parameter

due to the

coexistence of the tensile are evaluated:
(2-21)
where

and

are estimated based on the development of the fanning crack pattern along

the height of the plastic hinge region,
.

/

(2-22)

(2-23)
The influence of the concrete softening parameter, which controls the decay of the
capacity curve with inelastic deformations, is discussed further in Chapter 6.
Discussion
Figure 2.4 shows the analytical force-displacement responses [19] of the test units,
with varied nominal concrete compressive strengths from 5 ksi to 20 ksi and the
corresponding web crushing capacity curves of Hines and Seible. The description of the test
unit is included in Section 3.2. It can be seen that the web crushing capacity is predicted to
increase dramatically with an increase of concrete strength and the capacity is properly
predicted to decay with inelastic deformations. It should be noted that the maximum stress
limit of

has been applied on the capacity curves. This threshold is much less

conservative than that proposed by Oesterle et al. and Paulay and Priestley.
31

Displacement (in.)
1

2

3

4

5

1200

=20 ksi

1000

Force [kN]

6

=15 ksi

=10 ksi

300
250
200

800

150

600
400

100

=5 ksi

Force (kips)

0

50

200
F-D Response

0
0

20

40

60

80

100

120

0

140

Displacement [mm]
Figure 2.4 Analytical force-displacement responses of the single walls with web crushing
capacity curves of Hines and Seible
2.5.4

Discussion on Truss Models
The truss analogy is a simple in concept and seems easy to be implemented.

However, the evaluation of compressive strength of the critical strut entails the evaluation of
the strength of the damaged concrete material. This requirement is quite complicated when
considering the cyclic loading effects since the diagonal struts will experience cracking in two
directions upon load reversal. Thus, the evaluation of the compressive strength of the struts
cannot be appropriate until the effect of cyclic loading and crack width are fully considered.
In view of these difficulties, ACI 318-08 considers the shear strength of structures to be
related to the square root of the f’c with an upper-bound limit. However, the degradation of
f’c under complicated stress demands and load reversals is essential to evaluate the web
crushing capacity of HSC structural walls.
32

2.6 Nonlinear Finite Element Modeling Method
2.6.1

Continuum Damage Plasticity model in ABAQUS
The general purpose finite element program ABAQUS [35] was chosen in this study

for conducting three-dimensional (3D) modeling due to its robustness to simulate nonlinear
mechanics problems and its distinctive concrete material models. There are two main
concrete models in the material library of ABAQUS, one is the concrete smeared cracking
model; the other is the concrete damaged plasticity (CDP) model. The concrete smeared
cracking model is mainly designed for monotonic loading and may not be applied for the
walls tested under cyclic loading. In contrast, the CDP model is regarded to be able to better
represent the behavior of concrete since it is well founded on plasticity theory with cracking
being regarded as a limit state of plastic yielding and various types of loading can be applied,
including cyclic loading. With the introduction of concepts on damage mechanics, stiffness
degradation can be considered through the definition of damage indices.
The features of the CDP model are described in the following. Damage is introduced
in the definition of the elastic modulus, and the tensile and compressive plastic strains to
represent the inelastic behavior of concrete. Two different damage indices are required to
define cracking in tension and compression separately. The CDP model can be used for
cyclic loading with the unilateral effect being explicitly considered, i.e., the compressive
stiffness is recovered upon crack closure as load changes from tension to compression. A
non-associated flow rule is used to derive the plastic strain rate since the associative flow rule
could result in the problem in the control of dilatancy.
Overview of the CDP model

33

The framework of the CDP model is described briefly in the following. For a rateindependent model, the total strain rate ̇ can be decomposed into an elastic part ̇

and a

plastic part ̇
̇

̇

(2-24)
̇

The constitutive equation of the material with scalar isotropic damage is expressed as
(
where

is the Cauchy stress tensor,

variable,

is the strain tensor,

)

(

)

(2-25)

is the corresponding scalar stiffness degradation

is the initial (undamaged) elastic stiffness, and

is the degraded elastic stiffness tensor.
Based on the concepts of continuum damage mechanics, the effective stress tensor is
defined as
̅
where

(

)

(2-26)

is the plastic strain tensor. The Cauchy stress is related to the effective stress

through a scalar damage index, , by:
̅

(2-27)

Hardening variables
Damage states in tension and compression are characterized independently by two
hardening variables ̃

and ̃

, which are referred to as the equivalent plastic strains in

tension and compression, respectively. The evolution of the hardening variables is expressed
as
̃

* ̃

34

̃ +

(2-28)

̇̃

̅ ̃

(2-29)
̇

Yield function
The yield function,

( ̅ ̃ ), represents a surface in the effective stress space,

which determines the states of failure and damage. For the inviscid plastic-damage model
(̅ ̃ )

(2-30)

The CDP model in ABAQUS uses the yield function proposed by Lubliner et al. [36]
and incorporates the modifications proposed by Lee and Fenves [37] to account for different
evolution of strength under tension and compression. In terms of effective stress, the yield
function takes the form
(̅
where ̅

〉

̂
〈 ̅

̅ , is the effective hydrostatic pressure; ̅

equivalent effective stress; ̅
̂
and ̅

̂
( ̃ )〈̅
̅

̅

〉)
√

(2-31)

̅ ̅ , is the Mises

̅ , is the deviatoric part of the effective stress tensor ̅;

is the maximum eigenvalue of ̅. The coefficients of , , and
(

)

(

)

̅ ( ̃ )

are defined by:
(2-32)

(2-33)

̅ ( ̃ )

(2-34)
where

is the ratio of initial equibiaxial compressive yield stress to uniaxial

compressive yield stress;

is the ratio of the second stress invariant on the tensile meridian

35

to that on the compressive meridian; ̅ ( ̃ ) is the effective tensile stress; and ̅ ( ̃ ) is
the effective compressive stress.
Flow rule
Plastic flow is governed by a flow potential
̇

̇

according to the flow rule
̅

where ̇ is the nonnegative plastic multiplier. ̇ and
̇

; ̇

;

(2-35)

̅

obey the Kuhn-Tucker conditions:

. The plastic potential is defined in the effective stress space.

A non-associated plastic flow is used in the CDP model. The flow potential

is the

Druckder-Prager hyperbolic function
√(
where

)

is the dilation angle measured in the ̅

the uniaxial tensile stress at failure; and

̅

(2-36)
̅

̅ plane at high confining pressure;

is

is the flow potential eccentricity. A result of

introducing a non-associated flow rule is that the stiffness matrix will be asymmetric. In
ABAQUS, an asymmetric matrix storage and solution scheme can be used to improve
computational efficiency.
Damage and stiffness degradation
The evolution equations of the hardening variables ̃

and ̃

are conveniently

formulated by considering uniaxial loading conditions first and then extended to multi-axial
conditions.
Uniaxial conditions
As shown in Figure 2.5 and Figure 2.6, when a concrete specimen is unloaded from
any point on the strain softening branch, the unloading stiffness degrades. The degraded
36

response of concrete is characterized by two independent uniaxial damage variables,

and

, which are assumed to be functions of the plastic strains by neglecting the factors of
temperature or other field variables
( ̃ )
(2-37)
( ̃ )
The uniaxial degradation variables are increasing functions of the equivalent plastic strains
which ranges from zero for undamaged material to one for fully damaged material.
The effective uniaxial cohesion stresses, ̅ and ̅ , are expressed as
̅

(

̃ )
(2-38)

̅

(

̃ )

Under uniaxial cyclic loading conditions, the degradation mechanisms are quite
complex, involving the opening and closing of the previously formed micro-cracks, as well
as their interaction. It has been observed through experiments that there is some stiffness
recovery under load reversal. This effect is more pronounced when tensile cracks close and
concrete is in compression. For uniaxial cyclic conditions, ABAQUS assumes that
(2-39)
where

and

are functions of stress state introduced to represent stiffness recovery

effects
̅
(

̅

where

37

)

(2-40)

̅

̅

{

̅
̅

̃
Figure 2.5 Concrete stress-strain curve under uniaxial tension

̃
Figure 2.6 Concrete stress-strain curve under uniaxial compression

38

(2-41)

The weight factors

and

control the recovery of the tensile and compressive

stiffness upon load reversal. Figure 2.7 illustrates the effect of the compression stiffness
recovery parameter

.

=1
=0
Figure 2.7 Illustration of the effect of the compression stiffness recovery parameter

.

Multiaxial conditions
Based on the work of Lee and Fenves [37], the equivalent plastic strain rates are
evaluated according to
̂
̅ ̂̇
̇̃

(2-42)
̇̃
where ̂̇
tensor ̇

and ̂̇

̂
̅ ) ̂̇

(

are the maximum and minimum eigenvalues of the plastic strain rate

and

39

∑

̂
〈̅ 〉

∑

̂
̅

̂
|̅ |

̂
̅

(2-43)

̂
is a stress weight factor that is equal to one (1.0) if all principal stresses ̅ ,

, 2, 3 are

positive and equal to zero (0.0) if they are negative. The Macauley bracket 〈 〉 is defined by
〈 〉

| |

.

The evolution equation for general multiaxial stress conditions can be expressed in
matrix form as
̇

* ̇̃

̇̃ +

̂ ( ̅ ̃ ) ̂̇
̂

(2-44)

where
̂ (̅ ̃ )
̂

0

̂̇

̂̇

̂
̅

1
̂
̅ )

(

(2-45)

̂̇

(2-46)

and
[ ̂̇

̂̇ ]

̂̇

̂̇

The definition of the scalar degradation variable

is consistent with the uniaxial

conditions with equations shown in Equation (2-39) and (2-40).
Viscoplastic regularization
The softening behavior of concrete under both tension and compression leads to
severe convergence difficulties. The availability of a viscoplastic regularization in ABAQUS
Standard permits overcoming these difficulties. ABAQUS uses a generalization of the
Duvaut-Lions regularization method, according to which the viscoplastic strain rate tensor,
̇ , is defined as
̇

(

40

)

(2-47)

where

is the viscosity parameter representing the relaxation time of the viscoplastic system.

Similarly, a viscous stiffness degradation variable,

, is defined as

̇
where

(2-48)

is the degradation variable evaluated in the inviscid backbond model. The stress-

strain relation of the viscoplastic model is given as
(

)

(2-49)

Integration of the CDP model
The model is integrated using the backward Euler method generally used with the
plasticity models in ABAQUS [35][38]. A material Jacobian consistent with this integration
operator is used for the equilibrium iterations.
The description of damage and the definition of damage evolution curve for the
material input of the model parameter is presented in Chapter 5.
2.6.2

MCFT based FE modeling – VecTor2

Overview
2D plane stress finite element analysis can be implemented at a lower computational
cost to simulate the response of reinforced concrete beams and shear walls under in-plane
loading. VecTor2 is a program based on the Modified Compression Field Theory (MCFT)
[28] for nonlinear finite element analysis of reinforced concrete membrane structures [39]. It
incorporates a variety of phenomenological nonlinear material models into the framework of
the finite element method to represent the realistic features of the behavior of reinforced
concrete structures observed from experiments, e.g., compression softening, tension
stiffening and tension softening, etc.

41

Theories of VecTor2 – MCFT and DSFM
Two theories/models form the theoretical core of VecTor2, the modified
compression field theory (MCFT) and the disturbed stress filed model (DSFM). MCFT was
proposed based on tests of reinforced concrete membrane elements subjected to shear and
normal stresses. The outline of MCFT is shown in Figure 2.8 [40]. The theory is composed
of three sets of relationships: equilibrium equations with average stresses in concrete and
reinforcement; geometrical compatibility conditions; and the average stress-average strain
constitutive relationships of cracked concrete and reinforcement. The first two relationships
are derived based on Mohr‟s circle of stress and strain. MCFT is based on the smeared
rotational crack concept, which was verified through membrane tests.

Figure 2.8 The modified compression field theory (MCFT) [40].
42

The DSFM [39] addresses the deficiencies of the MCFT in predicting the response
of certain structures and loading scenarios. The DSFM is conceptually similar to the MCFT,
but extends it in several aspects. Most importantly, the DSFM augments the compatibility
relationships of the MCFT to include crack shear slip deformations. The strains due to the
slip deformations are distinguished from the strains of the concrete continuum due to stress.
Therefore, the DSFM decouples the orientation of the principal stress field from that of the
principal strain field, resulting in a smeared delayed rotating-crack model.
Analysis features
Some other features of VecTor2 include a comprehensive set of so-called
phenomenological constitutive models to consider both major and secondary effects in the
response of reinforced concrete structures. An important feature, which is essential for the
single wall analyses in this study, is the concrete strength increase due to the lateral
confinement provided by transverse stirrups, which can be adequately captured by VecTor2.
Modeling of triaxial stresses
Although the MCFT is formulated for a state of plane stress, VecTor2 accounts for
out-of-plane stresses (z-direction) due to the confinement of lateral concrete expansion by
out-of-plane reinforcement. A triaxial stress state is then utilized in computing the strength
enhancement due to confinement. The out-of-plane concrete strain is computed as follows.
(
where

̅

)

̅

(2-50)

is the reinforcement ratio of the out-of-plane reinforcement. If the out-of-plane

reinforcement has yielded, the out-of-plane concrete strain is computed as follows
̅

The stress,

̅

, in the out-of-plane reinforcement is determined as follows.
43

(2-51)

(2-52)
The resulting out-of-plane concrete compressive stress,

, is determined from equilibrium

as follows.
(2-53)
Finite element implementation
Regarding the finite element implementation in VecTor2, reinforced concrete is dealt
as an orthotropic material in the principal stress directions. The material secant stiffness
matrix of concrete is expresses as follows by ignoring the Poisson‟s effect,
̅
[

]

̅

6

7

(2-54)

̅
The secant moduli, ̅

, ̅

, and ̅ are calculated as follows,

̅

̅

; ̅

; ̅

̅

̅
̅

(2-55)

The material secant stiffness matrix of i-th reinforcement component based on its
longitudinal axis is calculated as follows,
̅
[
where

]

[

]

(2-56)

is the reinforcement ratio of the corresponding component. The secant modulus

̅ , is calculated as ̅

.

The global stiffness matrix is assembled in the global coordinate system as follows,
[ ]

[

]

where,

44

∑

[

]

(2-57)

[
[

[ ]

[ ] [

]

][ ]

(2-58)

[ ] [

]

][ ]

(2-59)

[

]

(2-60)

The constitutive equation of stress and total strain is expressed as
[ ]
[

[ ][ ]

[

]

(2-61)

] is a pseudo-stress produced due to strain offsets in concrete and reinforcement,

defined as
[
where *

]

[

] ,[

]

*

+

*

+-

∑

[

] ,*

+ are the elastic strain offsets in concrete, *

concrete, and [ ] are the strain due to crack shear slip. *
concrete and *

+

*

+-

(2-62)

+ are the plastic strain offsets in
+ are the elastic strain offsets in

+ are the plastic strain offsets in concrete.

2.7 Research Hypothesis of This Study
Evidence supporting the hypothesis that HSC increases the web crushing capacity of
structural walls has been available for some time. Intuitively, web crushing strength should
be proportional to the concrete compressive strength in view of the fact that the struts
fracture under compression. Research by the Portland Cement Association (PCA) in the
1970‟s on walls with boundary elements [5] noted that specimen B6 with a concrete
compressive strength of 22 MPa (3,165 psi) failed in web crushing at significantly lower
deformation capacity than specimen B7 with a concrete compressive strength of 49 MPa
(7,155 psi). However, no further HSC structural walls were tested.
45

All the reviewed models in Section 2.5 show that web crushing strength is linearly
related to concrete compressive strength, indicative of new possibilities for increasing the
web crushing capacity of slender structural members with increased concrete compressive
strength. However, this potential is currently impaired by outdated design provisions and
lack of experience. The lack of experimental data has been dealt with by setting an upperbound limit as noted in Equation (2-15) and Equation (2-16). However, without careful
calibrations with experimental data from structural wall tests with various concrete strengths,
the specified values are questionable. While there is no explicit upper-bound limit in the
model by Hines and Seible [16], its reliability to predict capacities for HSC structural walls is
expected to deteriorate since the model was calibrated with data from tests of normalstrength-concrete walls. The ACI 318-08 [25] code does not reflect the direct dependence of
web crushing capacity on

, but rather limits shear demand based on √ , a quantity

commonly related to concrete tensile strength and diagonal tension failure of structural walls.
It should be noted that a maximum value of 100 psi [0.69 MPa] has been applied by the code
because of the lack of test data and practical experience with concrete strength over 10,000
psi (69 MPa).
The research summarized in this dissertation was thus developed with the intention
of testing the following hypotheses:
1. Web crushing strength increases in proportion to f’c as long as the struts are not
damaged. Hence, transformation from an elastic web crushing failure to an inelastic
web crushing failure can be achieved simply by increasing the concrete strength.
2. Damage to struts caused by multi-directional loading, cyclic loading and inelastic
deformations can limit web strength independently of f’c. Hence, increases in f’c may
not lead to proportional increases in ductility capacity.
46

3. The contribution of cyclic loading to strut degradation can be observed by
comparing test units loaded cyclically to similar test units loaded monotonically.
4. Three-dimensional shear demands on wall assemblies curtail web crushing capacity
by virtue of a reduced strut geometry and combined flexure/shear stress demand.
Nonetheless, acceptable predictions for three-dimensional wall assemblies may be
obtained by resolving the flexure and shear demands into a two-dimensional model.
5. Inelastic web crushing capacity can be predicted with sufficient accuracy to support
the design of thin webbed elements that experience “ductile shear failures” which
may be easier to repair than ductile flexural failures. Inelastic shear behavior and
failure can thus be an acceptable and reliable failure mode for seismic performance
design.
6. Concrete strengths for high-strength-concrete can be reliably obtained to facilitate
such designs.
7. High-strength-concrete can transform the design of pier walls and other slender
reinforced concrete elements under seismic loads.

47

Chapter 3.
3.1

Experimental Studies on HSC Structural Walls

Summary
A common preconceived notion is that use of high-strength-concrete (HSC) can

jeopardize the ductility level of the structures in seismic regions. However, a well-designed
structure with appropriate detailing may take full advantage of the benefits of HSC with
adequate ductility being achieved. In this study, the seismic performance of HSC structural
walls with highly-reinforced boundary elements and thin constrained web was experimentally
evaluated. Eight 1/5-scale single walls were tested with design concrete compressive
strengths of 34, 69, 103, and 138 MPa (5, 10, 15 and 20 ksi) under cyclic and monotonic
loading protocols. Among the possibilities of various shear failure mechanisms, web
crushing is the only failure mode designed to happen. The objective of this study was to
evaluate the effect of concrete strength on the web crushing performance limits of HSC
structural walls. The web crushing behavior of the structural walls with same design concrete
compressive strength but under different loading protocols, monotonic and cyclic loading,
was compared to assess the effect of loading protocol and damage accumulation on web
crushing capacity. A new concept of “ductile shear failure” is introduced in contrast to the
conventional and inferior inelastic shear failure mechanism. The experimental results
revealed that high-strength-concrete can effectively delay web crushing failures and increase
the displacement ductility capacity of structural walls. However, web crushing capacity can
be significantly curtailed under cyclic loading, particularly as the concrete compressive
strength increases. This is attributed to the rapid shear strength and stiffness degradation
under cyclic loading. Nonetheless, ductile shear failures were still obtained for HSC
structural walls up to moderate displacement ductility levels.
48

3.2 Experimental Investigation
The structural walls featured highly-reinforced boundary elements and a thin
constrained wall web. An aspect ratio of 2.5 was maintained, which falls at the interface of
slender and squat walls. The cross section of the walls resembles a dumbbell shape with two
highly-reinforced boundary elements at the ends of the thin wall web. The boundary
elements are designed to ensure that the moment capacity of the walls is above the shear
capacity. The walls were designed such that the shear failure mechanism was web crushing.
Thus, diagonal tension failure of the wall under shear was avoided by the configuration of
the wall transverse reinforcement. It should be noted that researchers have reported that the
introduction of diagonal reinforcement can dramatically increase the diagonal tension
capacity in walls. Further research on this detailing aspect is out of the scope of this study
[41] and thus not considered. The constraint provided by the boundary elements along the
edge of the web prevents instability of the walls. The extension of the boundary element
steel cages into the footing provides enough anchorage to prevent sliding failure of the walls.
The concrete in the boundary elements is thus expected to display enhanced strength and
ductile behavior due to the confinement provided by the stirrups and the footing. The welldesigned structure will ensure the development of a plastic region under flexure. At the same
time, flexural deformations are introduced into the wall web. The mixed flexure and shear
demand and its effect on the web crushing capacity and ductility level is the foci of this
study.
This research examines the use of high-strength-concrete (HSC) to obtain ductile
shear failure behavior and to improve the seismic performance of structural walls. The effect
of concrete strength on web crushing failure mechanism, strength, and ductility has been

49

studied. The approach is non-intuitive as web crushing is a brittle failure and it is well known
that concrete becomes more brittle as its compressive strength increases. Comparative
results have been obtained to assess the cyclic loading effect on concrete damage, structural
strength/stiffness degradation, and web crushing capacity. The appropriate performance
evaluation on HSC structural walls exhibiting ductile shear failures is expected to contribute
to the groundwork of the next stage in earthquake resistant design of thin-webbed elements
and systems.
3.2.1

Single Wall Test Units
To verify the above-noted hypothesis and establish rational performance levels on

the inelastic web crushing limits for HSC structural walls, eight 1/5-scale cantilever structural
walls with design concrete compressive strengths of 34, 69, 103, and 138 MPa (5, 10, 15 and
20 ksi) were tested under cyclic and monotonic loading [8]. The walls consisted of highlyreinforced boundary elements with constrained thin web which were designed to induce the
desired web crushing failure mode and were not intent to represent a component from a
prototype structure. The test unit cross sections with reinforcement details are shown in
Figure 3.1. The identification name for the test units starts with „M‟ followed by two digits
2

2

denoting the design concrete compressive strength in kips/in (1 kip/in (ksi) = 6.895 MPa)
and then by a letter describing loading protocol: „C‟ for cyclic and „M‟ for monotonic
loading. The structural walls have a height of 2540 mm (100 in.) from the base up to the
mid-height of the loading block, which results in an aspect ratio of 2.5. As is shown in Figure
3.1, the steel reinforcement was essentially the same with a small variation for test unit
M15C. The change was made to facilitate specimen construction while a close reinforcement
ratio was maintained. The wall transverse steel is spaced at 76 mm (3 in.) vertically for walls
50

M20M and M20C while the spacing is 102 mm (4 in.) for the rest. The transverse steel was
designed to avoid diagonal tension failure.
The walls were loaded monotonically and cyclically according to an, incrementally
increasing fully-reversed cyclic pattern with constant axial load. The axial load for all test
units was 579 kN (130 kips), corresponding to

for a compressive strength of 34

MPa (5 ksi). The axial load was applied by means of hydraulic jacks and high-strength rods
reacting against the wall top load stub through a spandrel beam. The horizontal load was
applied with a servo-controlled actuator connected to a load stub at the top of the wall.
Lateral stability was provided by means of a pair of parallel inclined tensioned chains on
both side of the wall. An overview of the test setup is shown in Figure 3.2.

51

1016
(40")
254
(10")

254
(10")

508
(20")

76
(3")

254
(10")
76
(3")

254
(10")

254
(10")

13 (1/2")
cover

M22 (#7)
4 mid-points
M25 (#8)
4 corners
M10 (#3)
stirrup
76 (3") o.c.

15C

M10 (#3)
127 (5") o.c.

M10 (#3)
76 (3") o.c. for M20C and
M20M; 102 (4") o.c. for
the rest

Cross section of M05C, units M05C, M05M, M10C,
Test Units the test M05M, M10C, M10M,
M10M, M15M, M20C, and M20M.
M15M, M20C, M20M
1016
(40")
254
(10")

508
(20")

254
(10")

254
(10")

76
(3")
254
(10")

13 (1/2")
cover
M19 (#6)
12 tot.
M10 (#3)
stirrup
102 (4") o.c.

13 (1/2")
cover

M22 (#7)
4 mid-points
M25 (#8)
4 corners
M10 (#3)
stirrup
76 (3") o.c.

M10 (#3)
102 (4") o.c.
M10 (#3)
127 (5") o.c.

Cross Test Unitthe test units M15C.
section of M15C
Figure 3.1 Single wall test unit cross sections with reinforcement details.

52

M1
127 (5

Te

Figure 3.2 Single wall test setup overview.
(For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.)
53

3.2.2

Material Properties
Modern concrete technology makes it realistic to produce 138 MPa (20,000 psi) or

even higher strength of concrete with traditional mixing ingredients and without using any
exotic aggregates or special process. The addition of silica fume to obtain HSC is essential
because its high fineness property improves the hydration process. Concrete mixture
proportions for all the test units are shown in Table 3-1. The concrete compressive strength
development was monitored by a series of cylinder tests for each batch of concrete. The
corresponding concrete strength development curves are shown in Figure 3.3. The concrete
material properties of compressive strength, tensile strength and modulus of rupture were
evaluated at day-of-test. The data is shown in Table 3-2.
Table 3-1 Summary of the concrete mixtures for the single wall test units.
Item

M05C
M05M

M10C

M10M

M15C
M15M

M20C
M20M

Cement Type I, kg

167

167

345

393

431

Fly ash, kg

89.4

89.4

–

–

–

Silica fume, kg

–

–

–

32.7

68.1

Fine Aggregate, kg

695

695

619

524

454

Coarse Aggregate, kg

795

795

788

826

840

HRWR, Type F, liter

1.00

1.00

9.47

12.0

6.51

Retarder, Type D, liter

–

–

–

–

1.30

Air, liter

–

–

–

0.622

–

Water, kg

116

116

121

99.4

106

Water-cementitious ratio

0.45

0.45

0.35

0.25

0.21

Note: 1 lb = 0.454 kg; 1 oz = 0.0296 liter.

54

90

75
60

15.0

45

12.5

30

10.0

15
0

45

0

30

14

14

28

42

60
30
15
0
14

70 5.084

Days

0

45

4217.5 56

2.5

M05C
12.5
M05M 84
56
70
98
10.0
M10C
Days M10M
7.5
M15C
M15M 5.0
M20C
2.5
M20M

0.0
112

15.0

0

0

28

15

75

7.5

20.0

M05C
M05M
M10C
M10M
M15C
28
42
56
70
M15M
Days
M20C
M20M
M05C

84

98

0.0
112

Figure 3.3 Concrete material strength development curves for single wall tests.
M05M
M10C
Table 3-2 Concrete material M10M at day-of-test: Compressive strength, tensile strength
properties
M15C rupture for the single wall test units.
and modulus of
M15M
Strength
M20C
M05C M05M M10C M10M M15C M15M M20C M20M
(MPa)
M20M
46.0
38.9
56.4
84.0
102
111
131
115
dev.

1.86

1.37

1.01

4.99

3.01

2.55

3.55

4.50

5.54

5.70

6.17

6.19

5.96

0.207

0.0966

0.331

0.490

0.441

0.910

0.400

0.172

5.49
dev.

1.23

3.25
dev.

1.37

5.37

6.57

7.33

9.01

9.35

11.5

10.2

0.0897

0.0551

0.221

0.669

0.559

1.08

0.359

0.593

Note: 1 MPa = 0.145 ksi.

Table 3-3 lists the steel reinforcement properties for all the single wall test units and
the corresponding modeling parameters. Most of the rebars displayed an obvious yield
plateau and were fitted with Mander‟s model (M) [42], except for two #8 rebars without

55

Concrete compressive strengt

105

90

60

Concrete compressive strength (ksi)

120

Concrete compre

Concrete compressive strength

Concrete compressive strength [MPa]

135

105

9

obvious yielding are fitted with Menegotto-pinto model (MP) [43]. Limited by the capacity of
the MTS Grip System used for rebar test, the #8 rebars could be tested until fracture and
only a small range of the stress-strain curves was obtained.
Table 3-3 Steel properties and modeling parameters for the single wall test units.
Test Unit

Model

Size

459

703

0.0075

7759

464

697

0.0096

8966

M22

448

672

0.0094

8966

M10

476

746

0.0060

8966

545

730

0.0050

7586

439

705

0.0078

8966

M10

481

759

0.0054

9655

503

756

0.0030

7931

421

630

0.0093

8621

M10

478

748

0.0060

10690

510

656

0.0072

5517

451

703

0.0054

9310

M22

446

699

0.0060

8966

M10

438

703

0.0043

8793

M10

M

7759

M25
M20C
M20M

0.0083

M10

M

692

M22
M15M

445

M10

M

M10

M19
M15C

0.0026

M10

M

672

(MPa)
11262

M10

M10C
M10M

M22
M

(MPa)

M25

M05C
M05M

(MPa)
448

443

717

0.0037

8621

–

r

–

2.5

(ksi)
2759

–

6

11034

Test Unit

Model

Size

M15M
M05C
M05M

MP

M25

(ksi)
586

MP

M25

524

Note: 1 MPa = 0.145 ksi.

56

3.2.3

Instrumentation
The test units were instrumented to measure flexural and shear deformations,

flexural curvatures, and steel strains at specific locations in the steel reinforcement. Flexural
deformations were obtained by the integrating the flexural curvatures at the extreme edges of
the boundary elements. Flexure curvatures were calculated from displacement transducer
measurements mounted on both tension and compression sides of the boundary elements
along the wall height. Shear deformations were determined by using two instrumentation
panels featuring two independent displacement transducers arranged diagonally on the south
wall face. The global displacement was measured at the center of the west surface of the load
stub with a displacement transducer. The contribution to lateral deformations from strain
penetration, or bond-slip, effects into the footing block was monitored by measuring
deformations of a rigid reference frame (cantilevers anchored on the footing surface to avoid
the possible disturbance of the footing concrete cracking) with respect to the footing top
over a gage length of 51 mm [2 in.]. Strain gauges were installed at strategic locations of the
longitudinal and transverse steel reinforcement to monitor flexural yielding of the
longitudinal reinforcement in the boundary elements and yielding of the transverse
reinforcement in the wall web. The instrumentation layout for LVDTs and string pots is
shown in Figure 3.4.
3.2.4

Loading protocol
Cyclic and monotonic loading was applied on the two test units of equal design

concrete strength, separately, to evaluate the effect of loading history on the shear
strength/stiffness degradation and displacement limits. Loading was applied by means of a
servo-controlled hydraulic actuator mounted on a reaction frame and attached to the test
57

unit through a stiff top block. The cyclic loading protocol is shown in Figure 3.5. Four initial
cycles were applied in force control until the theoretical first yield force,

, the force at

which the extreme steel rebar in tension first yields, was reached. The remainder of the test
was conducted in displacement control with two cycles at each displacement ductility levels
= 1, 1.5, 2, 3, 4 and 6, or until failure of the test unit. The ideal yield displacement,
defined as displacement ductility one,

= 1, was derived based on the stiffness at first yield

and the theoretical force at which either the extreme confined concrete fibers reached
0.004 or the extreme steel fiber in tension reached
applied in force control until
and

,

=

= 0.015. The monotonic loading was

and then in displacement control until failure. The values of

which are needed to define the loading protocol are listed in Table 3-4.

58

1219

P

229

F

Global_disp

LCA88

LCG88
457

LCA70

LCG70
457
e
Sh

ar_

LCG52

4
sp
Di

2540

Sh
ea
r_
Di
sp

3

Shear_Disp3 Shear_Disp4

LCA52

LCA40

LCG40
914

LCA28

LCG28

LCA16

LCG16

25

LCG10
LCG6
LCG2

2
isp

Sh

LCA6
LCA2

r_
ea

D

LCA10

r_
ea

Sh

1
isp
Shear_Disp1 Shear_Disp2
D

51

25

914

Dimension unit in millimeter.
1 in. = 25.4 mm.

1829

Figure 3.4 Single wall test unit instrumentation layout.

59

152
102
102
51

Cycles =
4

2

2

2

2

2

2

8

 = 6

Displacement Ductility  

6

load control
4 cycles

4

 = 3

1/4 Fy
3/4 Fy

2

 = 2

  = 1.5

 = 1

 = 4

0
-2
Fy

-4

1/2 Fy

-6

 =   y

displacement control

-8
0

1

2

3

4

5

6

7

8

9

10

11

12

13 14

15

16

No. of Cycles

Figure 3.5 Cyclic loading protocol for single wall tests.
Table 3-4 Values of force at first yield and ideal yield displacement for single wall tests.
Test

M05C

M05M

M10C

M10M

M15C

M15M

M20C

M20M

(kN)

578

576

583

618

586

644

576

572

(mm)

25.7

26.9

22.9

19.1

19.6

20.6

19.1

21.3

Note: 1 kip = 4.45 kN; 1 in. = 25.4 mm.

3.3 Experimental Observations
First, the common behavior observed on all test units is described. No cracks were
observed through the loading up to

. At

, flexural cracking in the boundary

elements and diagonal shear cracking in the webs appeared. At

, shear cracks were

observed to spread throughout the entire web. The development of cracking for the single
walls under cyclic loading at

is illustrated through Figure 3.6 and Figure 3.7. It can

60

be seen that HSC walls have relatively denser flexural and shear cracking than the NSC walls,
which complies with the brittle nature of HSC. Figure 3.8 shows the cracking of M05M at
. With the increase of displacement ductility, more cracks developed between the existing
cracks and the active cracks became wider. Overall, the spacing of shear cracks for the HSC
walls was much smaller than for the NSC walls. Even though the diagonal shear cracking
under tension does not control the capacity of the walls, it defines the height and width of
the struts formed by adjacent cracks and thus affects the struts compressive capacities.
Moreover, the width of the cracks affects the shear slip behavior at the crack interface,
which affects the compressive capacity of the strut as well. Another common feature in the
test units was the spalling of the cover concrete in the compression boundary element.
Relatively larger spalling was observed on the HSC wall units. Nonetheless, since the
columns are well-confined by the stirrups, the boundary elements had no problem resisting
the compression force well.
Second, the peculiarities of the web crushing failures and the displacement ductility
reached by the test units are described individually. Test unit M05C failed on the first
excursion to

= 2. Web crushing was observed to occur at

shown in Figure 3.9. Wall M10C failed on the first excursion to
observed to occur at

= 1.8. The failure mode is
= 2. Web crushing was

= 1.8. Upon failure, loading dropped approximately 40%,

demonstrating a significant loss of flexural strength. The failure mode is shown in Figure
3.10. Test unit M15C performed in a ductile manner up to

= 4. Web crushing was

observed to occur on the second excursion to ductility 4. The load dropped only
approximately 8%, compared with the corresponding load on the first cycle of ductility 4,
demonstrating that the flexural strength of the test unit did not deteriorate very much. The
failure mode is shown in Figure 3.11. Test unit M20C performed in a ductile manner up to
61

= 6. Web crushing was observed to occur on the first excursion to

= 6. Web crushing

started to happen at a displacement of 2.6 in. The strength of the test unit degraded around
40% when the displacement reached the target value for this cycle. The failure mode is
shown in Figure 3.12. Wall M05M failed at

= 2.3. The failure mode is shown in Figure

3.13. Test unit M10M performed in a ductile manner until web crushing occurred at

= 7.

The failure mode is shown in Figure 3.14. Wall M15M performed in a ductile manner until
web crushing occurred at

= 6.5. The failure mode is shown in Figure 3.15. Test unit

M20M performed in a ductile manner until web crushing occurred at

= 9.2. The failure

mode is shown in Figure 3.16.
All walls failed in web crushing according to the experimental aim. Test units with
lower concrete compressive strength (M05C, M05M and M10C) failed after only minor
levels of inelastic response (

). Tensile cracking was minimal and cracks fully closed upon

load reversal. No crack realignment was observed and thus these walls were limited by
standard, or elastic, web crushing. The failures were sudden and the crushing of the concrete
struts occurred along the interface of the wall web and the compression boundary element
and propagated along the wall height almost simultaneously.
The rest of the test units exhibited moderate to high ductile behavior before web
crushing failure. Cracking was much more extensive and crack spacing was much smaller.
The fanning flexure-shear cracking pattern was formed within the plastic hinge region with
fairly flat cracks close to the bottom and much steeper cracks at the top. For the walls tested
under cyclic loading, the crisscross cracking pattern under cyclic loading broke the concrete
in the wall web into small diamond-shaped blocks. Such damage was particularly evident in
the wall cover concrete. The excessive stress introduced by crack misalignment, shear
friction and distortion caused the web cover concrete to lose its bond to the reinforcement
62

and spall off. The test units gradually lost their load-carrying capacity as a result diminished
load transfer efficiency of the concrete struts. Web crushing was observed to expand over a
large area within the plastic hinge region of the wall and crushing of the flexure-shear struts
initiated in the center of the web and then rapidly extended to the edge of the compression
boundary element. Under monotonic loading, the compression struts remained integral,
though severely cracked, and therefore the test units were able to sustain larger inelastic
deformations, a difference that became more significant for higher values of concrete
compressive strength.

63

Figure 3.6 Test Unit M05C: south face,

= 1 x +1, push (positive) towards west.
64

Figure 3.7 Test Unit M20C: south face,

= 1 x +1, push (positive) towards west.
65

Figure 3.8 Test Unit M05M: south face, force control
66

, push (positive) towards west.

Figure 3.9 Test Unit M05C: north face,

= 1.8 x +1, push (positive) towards west.
67

M10C

Figure 3.10 Test Unit M10C: north face,

= 2 x +1, push (positive) towards west.
68

M15C

Figure 3.11 Test Unit M15C: north face,

= 4 x +2, push (positive) towards west.
69

Figure 3.12 Test Unit M20C: north face,
70

= 6 x 1, push (positive) towards west.

Figure 3.13 Test Unit M05M: north face,
71

= 2.3, push (positive) towards west.

Figure 3.14 Test Unit M10M: north face,
72

= 7, push towards west.

Figure 3.15 Test Unit M15M: north face,
73

= 6.5, push towards west.

Figure 3.16 Test Unit M20M: south face,
74

= 9.2, push towards west.

3.4 Experimental Results
The hysteretic force-displacement response of the four walls under cyclic loading is
shown in Figure 3.17 to Figure 3.20. Test data reduction was conducted to separate the
flexure and shear deformations in order to study the strength/stiffness degradation and
energy-dissipating capacity of the test units. Table 3-5 shows the calculated flexural and shear
deformations, their corresponding ratio and the calculated and measured total deformation
versus ductility levels for all walls. A comparison of the sum of the flexure and shear
deformations with the measured values verifies the accuracy of the curvature and shear
instrumentation. The ratio of the flexural and shear deformation ranges between 18~34%
with a mean value of approximately 24%, which was the value used to estimate shear
deformations in the force-displacement characterization described in Section 2.2.
Furthermore, it can be seen through the comparison of flexural and shear deformations that
the single walls exhibited a flexure-dominated behavior, which can also be seen in Figure
3.21 to Figure 3.24 where the hysteretic loops of M05C and M20C in flexure and shear are
shown. The hysteretic loops of the flexural deformation versus shear deformation show that
a constant ratio of the shear over flexural deformation could represent the relationship well,
as is illustrated in Figure 3.25, particularly at moderate to large levels of inelastic response.
Even if shear deformation is only a small portion of the total deformation, it cannot
be overlooked considering that web crushing is a shear failure mode. The contribution of the
shear deformation from the bottom panel and top panel was thus investigated. The bottom
panel mainly covers the plastic hinge region and is the area where web crushing initiates.
Refer to Figure 3.4 for the layout of the instrumentation. Figure 3.26 and Figure 3.27 show a
comparison of the shear behavior measured from the two panels for M05C and M20C,
respectively. It can be seen that comparable shear deformations were obtained from the top
75

and bottom panels for wall M05C. Almost a linear shear force-displacement relationship is
observed from the two panels. In contrast, the shear deformation from the bottom panel is
much larger than that from the top panel for wall M20C. A ductile shear behavior can be
observed on the bottom panel, while an elastic shear response is observed on the top one.
Based on the difference of the measured shear force-deformation response between the
NSC and HSC structural walls, the web crushing of NSC walls can be defined as an elastic
shear failure while that of the HSC walls is better defined as an inelastic shear failure.

76

Drift
-3% h
800

-2% h

-1% h

0% h

1% h

2% h

3% h

Fy

M05C

600

200
150

Fy'

100

200

50

Ductility

0

-2

-200

-1

0

1

2

0
-50

-400

Force (kips)

Force [kN]

400

-100

-600

-Fy'

-800

-Fy
-75 -60 -45 -30 -15

0

-150
-200

15

30

45

60

75

Displacement [mm]
Figure 3.17 Test Unit M05C: Hysteretic loops.

Drift

800

-2% h

-1% h

M10C

0% h

1% h

2% h

3% h

Fy
Fy'

600

150
100

Force [kN]

400
200
0
-200

Ductility
-3

-2

-1

0

1

2

3

-400
-800

50
0
-50
-100

-Fy'
-Fy

-600

200

-150
-200

-75 -60 -45 -30 -15

0

15

30

45

60

Displacement [mm]
Figure 3.18 Test Unit M10C: Hysteretic loops.
77

75

Force (kips)

-3% h

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
M15C

Fy
Fy'

600

150
100

Force [kN]

400
200
0
-200

200

50

Ductility
-5

-4

-3

-2
-2

-1

0

1

2

3

4

5

-400

0
-50

Force (kips)

800

-100
-Fy'
-Fy

-600
-800

-150
-200

-100

-75

-50

-25

0

25

50

75

100

Displacement [mm]
Figure 3.19 Test Unit M15C: Hysteretic loops.

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
Fy

M20C

150

Fy'

600

100

Force [kN]

400
200
0
-200

200

Ductility
-5

-4

-3

-2
-2

-1

0

1

2

3

4

5

-400
-600

-Fy

-100

-75

-50

-25

0

-50

-150
-200
25

50

75

Displacement [mm]
Figure 3.20 Test Unit M20C: Hysteretic loops.
78

0

-100

-Fy'

-800

50

100

Force (kips)

800

Table 3-5 Single wall tests: flexural and shear deformations versus ductility levels and
comparison of the calculated and measured global displacement
⁄

Single Wall
Test Unit
M05C
M05M
M10C

M10M

M15C

M15M

M20C

M20M

1x1
1.5x 1
1
1.5
1x1
1.5 x 1
1
1.5
2
3
4
5
6
1x1
1.5 x 1
2x1
3x1
4x1
1
1.5
2
3
4
5
6
1
1.5
2
3
4
1
1.5
2
3
4
5
6
7
8

(mm)
18.5
28.4
22.6
34.3
18.0
27.7
13.7
21.6
30.2
46.7
63.0
79.2
95.3
14.5
22.6
30.5
47.0
62.5
15.0
23.4
32.3
48.5
65.8
82.8
99.3
14.5
23.1
30.7
47.0
62.0
15.5
24.6
33.8
52.3
70.4
87.9
106
125
140

(mm)
5.08
7.62
7.37
10.4
6.10
7.87
3.81
5.33
6.60
9.65
13.2
16.8
20.3
3.56
4.83
6.60
9.40
13.2
4.06
5.59
6.86
9.65
11.7
15.7
19.6
3.30
4.57
6.60
9.91
15.5
3.81
5.33
6.86
9.91
13.5
17.3
21.3
27.2
32.3

Note: 1 in. = 25.4 mm.
79

0.27
0.27
0.33
0.30
0.34
0.28
0.28
0.25
0.22
0.21
0.21
0.21
0.21
0.25
0.21
0.22
0.20
0.21
0.27
0.24
0.21
0.20
0.18
0.19
0.20
0.23
0.20
0.21
0.21
0.25
0.25
0.22
0.20
0.19
0.19
0.20
0.20
0.22
0.23

(mm)
23.6
36.1
30.0
44.7
24.1
35.6
17.5
26.9
36.8
56.4
76.2
96.0
116
18.0
27.4
37.1
56.4
75.7
19.1
29.0
39.1
58.2
77.5
98.6
119
17.8
27.7
37.3
56.9
77.5
19.3
30.0
40.6
62.2
83.8
105
127
153
172

(mm)
25.7
38.4
31.0
45.0
24.6
36.1
19.1
28.7
38.4
57.4
76.5
95.5
115
19.8
29.5
39.4
58.9
78.7
20.6
31.0
41.1
61.7
82.3
103
123
19.1
28.7
38.1
57.4
76.5
20.1
30.7
41.4
62.7
84.3
106
127
151
170

Drift
-3% h

-2% h

-1% h

0% h

800

2% h

3% h

Fy

600

1% h

Fy'

200
150
100

200

50
Ductility

0

-2

-200

-1

0

1

2

0
-50

-400

Force (kips)

Force [kN]

400

-100

-600

-Fy'

-800

-Fy
-75 -60 -45 -30 -15

0

-150
-200

15

30

45

60

75

Displacement [mm]
Figure 3.21 Test Unit M05C: Hysteretic loop in flexure.

Drift
-2% h

-1% h

0% h

800

2% h

3% h

Fy

600

1% h

Fy'

150
100

Force [kN]

400
200
0
-200

Ductiilty
-2

-1

200

0

1

2

50
0
-50

-400

-100

-600

-Fy'

-800

-Fy
-75 -60 -45 -30 -15

0

15

-150
-200
30

45

60

Displacement [mm]
Figure 3.22 Test Unit M05C: Hysteretic loop in shear.
80

75

Force (kips)

-3% h

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
800

Fy

600

Fy'

200
150
100

200

50

0
-200

Ductility
-5

-4

-3

-2

-1
-1

1

0

2

3

4

5

-400
-600

-Fy

-100

-75

-50

-25

0

-50
-100

-Fy'

-800

0

Force (kips)

Force [kN]

400

-150
-200
25

50

75

100

Displacement [mm]
Figure 3.23 Test Unit M20C: Hysteretic loop in flexure.

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
800

Fy

600

Fy'

150

Force [kN]

200
-200

Ductility
-5

-4

-3

-2

-1

0

1

2

3

4

5

-400
-600

-Fy

-100

-75

-50

-25

0

-50

-150
-200
25

50

75

Displacement [mm]
Figure 3.24 Test Unit M20C: Hysteretic loop in shear.
81

0

-100

-Fy'

-800

50

100

Force (kips)

100

400
0

200

Flexural Displacement (in.)
-4

-3

-2

-1

0

1

2

3

4

2

50
Shear Displacement [mm]

30

1

20
10
0

0

-10
-20

-1

-30

Shear Displacement (in.)

Shear/Flexural Displacement = 0.24

40

-40
-50
-100 -80 -60 -40 -20

-2
0

20

40

60

80 100

Flexural Displacement [mm]
Figure 3.25 Test Unit M20C, shear displacements as a function of the flexural displacements

Drift
-1% h

0% h

800

Fy

600

1% h

Fy'

200
150
100

200

50

0

0

-200

-50

-400

Force (kips)

Force [kN]

400

-100

-600

-Fy'

-800

-Fy
-25 -20 -15 -10

-5

0

5

Bottom Panel -150
Top Panel
-200
10 15 20 25

Displacement [mm]
Figure 3.26 Test Unit M05C: Comparison of the shear deformation between the bottom and
top panel.
82

Drift
-1% h

0% h

800

Fy

600

1% h

Fy'

150

Force [kN]

200
-200

Ductility
-1

0

1

-400
-600

-Fy'

-800

-Fy

-25 -20 -15 -10

-5

0

5

50
0
-50

Force (kips)

100

400
0

200

-100
Bottom Panel -150
Top Panel
-200
10 15 20 25

Displacement [mm]
Figure 3.27 Test Unit M20C: Comparison of the shear deformation between the bottom and
top panel.
The curvature profiles of walls M10 and M20 are shown in Figure 3.28 and Figure
3.29. It can be seen that the flexural deformation demands from the monotonic and cyclic
test units was very similar. For the cyclically loading test units that failed at low to medium
ductility levels, the curvature distribution along the height was almost linear until failure. For
the monotonic loaded walls that failed at high ductility levels, the plastic rotations were
mainly concentrated within 200 mm [8 in.] from the bottom of the wall. Figure 3.30
compares strain profiles on the transverse steel at different height levels for walls M10C and
M20C. The wall transverse steel yielded before failure in both tests, which impairs the
confining effect onto the web concrete and most importantly, causes the cover concrete of
the web spall off easier and thus degrades the compression capacity of the diagonal struts.

83

Height abo

Height above footing, (in.)

2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0

Height above footing, h [mm]

Height above footing, h [mm]

0.0

Hei

Height above

40
1000
400
800
30
200
600
0
20
400
0 5 10 15 20 25 30 35 40 45 50 55 60 65 7
-3 rad/in.)
Average curvature (x10
10
200
-6 rad/mm)
Average 0
curvature (x10-3 rad/in.) 0.6 Average curvature (x10 2.4 2.7 3.
0.0 0.3
0.9 1.2 1.5 1.8 2.1 0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
=1 1
0.2 0.4 0.6 0.8 1.0 2200 1.4 1.6 1.8 x2.0
1.2
90
2000
Average curvature (x10-6 rad/mm)
=1.5 1
y=0.000116xrad/in.
M10C 1800
M10M
80
=1
=1 x 1
1600
y
=1.5 70
=1.5
1400 x 1
Average curvature (x10-3 rad/in.)
=2
60
1200
=1

Height above footing, h (in.)

Height above footing, h [mm]

=5
Average curvature (x10-6 rad/mm)
=6
Figure 3.29 Test Units M20:=1 x 1
Curvature profiles (average of push and pull maximums).
=7
=1.5 x 1
84 =8
=2 x 1
=9
=3 x 1

Height above footing, h (in.)

Height above footing, h [mm]

0.0 0.3 0.6 1000 1.2 1.5 1.8 =3 2.4 2.7 3.0 3.3
0.9
2.1
=1.5
50
90
800
=4
2200
=2
40
600
80
2000
=5
=3
=0.000116 rad/in.
y
30
400
1800
70
=6
=4
200
20
1600
60
=5 0
1400
10
0 10 20 30 40 50 60 70 80 90 100 110 1
50
=6
1200
0
40
1000
Average curvature (x10-6 rad/mm)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
800
30
=1 x 1
-6 rad/mm)
600
Average curvature (x10
20
=1.5 x 1
400
Figure 3.28 Test Units M10: Curvature profiles (average of push and pull maximums).
10
=1 200
x1
=2 x 1
-3
0
0
Average
=1.5 x 1 curvature (x10 rad/in.)
=3 x 1
0 10 20 30 40 50 60 70 80 90 100 110 120
=4 1
0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 x 3.3
=1
90
Average curvature (x10-6 rad/mm)
2200
=1
=1.5
M20C M20M
80
2000
=1 x 1
=1.5
=2
y=0.000116 rad/in.
1800
70
=1.5 x 1
=2
=3
1600
60
=2 x 1
=3
=4
1400
=3 x 1
50
=4
1200
=5
=4 x 1
40
=5
1000
=6
=1
800
=6
30
600
=1.5
=7
20
400
=2
=8
10
200
=3
=9
0
0
=480 90 100 110 120
0 10 20 30 40 50 60 70

xx1
1

x1

xx2
2

x2

xx1
1

x1

xx2
2

x2

xx

x

xx

x

xx

x

xx

x

xx

x

Strain

Strain

Strain

Strain
Strain
Strain
Strain

Strain
Strain

0.005 h = 4" 0.005
0.005 h = 4"
0.005 h h = 12"
0.005 = 12"0.005 h = 12"
h = 4"
0.004
0.004
0.004
0.004
0.004
0.004
= 0.0027
y y = 0.0027 y = 0.0027 0.003
= 0.0022
0.003
0.003
0.003 y y = 0.0022 y = 0.0022
0.003
0.003
0.002
0.002
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-508 -254
-508 -254 0 0 254 5080
-508 508 -254 0-508 254 5080
0
254 508
-508 254 508 254 -508 -254
-254
-254
254 508
0.005
0.005 h = 24"
0.005
0.005
0.005
h = 24"
h = 24"
h h = 8"
= 8" 0.005 h = 8"
0.004
0.004
0.004
0.004
0.004
0.004
0.003 y y
y
0.003
0.003
0.003 y
0.003 y y 0.003
0.002
0.002
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-508 508 -254 0-508 254 5080
0
254 508
-508 -254
-254
254 508
-508 -254 0 0 254 5080
254 508 254 -508 -254
-508 -254
0.005 h h = 36"
0.005 = 36"0.005 h = 36"
0.005 h h = 12"
0.005 = 12"0.005
h = 12"
0.004
0.004
0.004
0.004
0.004
0.004
yy
y
0.003
0.003 y y 0.003
0.003
0.003
0.003
y
0.002
0.002
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-508508 -254 0-508 254 5080
0
254 508
-508 -254
-508 -254 0 0 254 5080
254 508 254 -508 -254
-254
254 508
-508 -254
Test Unit M20CTest Unit M20C
Test Unit M10CTest Unit M10C
Test Unit M20C
Test Unit M10C

Note: 1 in. = 25.4 mm
Figure 3.30 Test Units M10C and M20C: Strain profiles of transverse steel.

85

A comparison of the force-displacement envelopes for the cyclic and monotonic
tests is shown in Figure 3.31 to Figure 3.34. Walls M05C, M05M and M10C failed at a
displacement ductility of about 1.5. The other walls exhibited moderate to high ductility
before web crushing. Test units M15C and M20C achieved a displacement ductility of 4
while units M10M, M15M and M20M failed at a displacement ductility of 6~9. The results
clearly show that the higher compressive strength allowed the walls to sustain higher inelastic
deformation and a stable hysteretic ductile response. It can be seen in Figure 3.31 that cyclic
loading had no significant effect on the capacity of the 34 MPa [5 ksi] test unit. However,
cyclic loading, and its resulting degradation, had a growing effect on the web crushing limits
with increasing concrete compressive strength. Thus, the gains in forestalling web crushing
with higher concrete compressive strength were curtailed by the greater susceptibility of
higher strength concrete to damage under cyclic loading, as higher concrete compressive
strength results in more brittleness and less energy-dissipation capacity from the material.
This explains the reasons for the comparable ductility levels achieved by walls M15C and
M20C. Thus, the shear stiffness degradation and damage of HSC structural walls under load
reversals has to be evaluated appropriately. Nonetheless, comparison of the response of
walls M15M and M20M shows that M20M had a larger deformation capacity, which
supports the hypothesis that the higher concrete strength can increase the inelastic capacity
of structural walls with confined boundary elements.

86

Drift Ratio (%)
0

1

2

3

4

5

6

7

8

250

1000
200
150

600

100

400
200

Force (kips)

Force [kN]

800

50

M05C (fc'=6.4ksi)
M05M (fc'=5.65ksi)

0

0
0

20

40

60

80 100 120 140 160 180 200
Displacement [mm]

Figure 3.31 Test Units M05: Comparison of the force-displacement envelopes

Drift Ratio (%)
0

1

2

3

4

5

6

7

8

250

1000
200
150

600

100

400
200

M10C (fc'=8.17ksi)
M10M (fc'=12.5ksi)

0

Force (kips)

Force [kN]

800

50
0

0

20

40

60

80 100 120 140 160 180 200
Displacement [mm]

Figure 3.32 Test Units M10: Comparison of the force-displacement envelopes
87

Drift Ratio (%)
0

1

2

3

4

5

6

7

8

250

1000
200
150

600

100

400
200

Force (kips)

Force [kN]

800

50

M15C (fc'=14.74ksi)
M15M (fc'=16.1ksi)

0

0
0

20

40

60

80 100 120 140 160 180 200
Displacement [mm]

Figure 3.33 Test Units M15: Comparison of the force-displacement envelopes

Drift Ratio (%)
0

1

2

3

4

5

6

7

8

250

1000
200
150

600

100

400
200

M20C (fc'=19.0ksi)
M20M (fc'=16.75ksi)

0

Force (kips)

Force [kN]

800

50
0

0

20

40

60

80 100 120 140 160 180 200
Displacement [mm]

Figure 3.34 Test Units M20: Comparison of the force-displacement envelopes
88

3.5 Discussion
Table 3-6 compares the web crushing capacities of the eight single walls tested in the
program with the capacities predicted by different models. It can be noted that the ACI
shear provision considerably underestimates the web crushing strength. At the same time,
the prediction quality of the model by Hines and Seible [16] on cyclic tests deteriorates with
the increase of concrete compressive strength since the model was calibrated using wall tests
with normal-strength-concrete and the strength and stiffness degradation of HSC was not
taken into account. Thus, the experimental program revealed that rational web crushing
models like the one by Hines and Seible need further considerations to be applicable to HSC
structural walls.
The experimental program has clearly proven the hypothesis that HSC can forestall
web crushing failures and allow the system to attain larger levels of inelastic deformation,
thus obtaining an inelastic flexure-shear response, or a ductile shear failure. It is clear that the
shear stress demands on the walls are well in excess of currently prescribed limits. However,
the experiments revealed that while HSC enables the shear-carrying compressive struts to be
stronger, and thus sustain higher effective wall shear stresses, the effect damage
accumulation due to cyclic loading on HSC needs to be carefully evaluated. A preliminary
evaluation was done with the model by Park and Ang [44] to assess the damage on the test
units due to ultimate deformation and hysteretic energy dissipation. Figure 3.35 illustrates the
damage indices of the four cyclic test units. The tendency of the damage indices based on
test data can be fitted by an exponential function very well. It can be seen that with the
increase of concrete strength, the damage due to ultimate deformation decreases while that
due to the energy dissipation increases. Thus, the fact that HSC increases the energy
dissipation capacity and ductility of the structural walls has been verified through tests.
89

However, it can be noted that the damage indices remain relatively unchanged for walls
M15C and M20C, indicating that no significant additional inelastic deformation capacity was
gained by increasing the concrete strength between these two test units. This effect is
attributed to the decrease in fracture toughness of the concrete with compressive strength,
which curtails the capacity of the shear-carrying struts. Further studies on this aspect are
required and are considered fundamental in establishing quantitative limits to the inelastic
web crushing behavior of HSC walls evidenced in this study.

90

Table 3-6 Comparison of inelastic web-crushing capacities of single walls with shear models.
Test
Unit

Experiment

ACI (2008)

Oesterle et al. (1984)
(mm)

(kN)

Diff.
(%)

Paulay and Priestley
(1992)
Diff.
(mm) (kN) (%)

Hines and Seible
(2004)
Diff.
(%)
(mm) (kN)

(mm)

(kN)

Diff.
(%)

8.64

342

81

32.8

765

27

23.6

714

47

48.5

821

8

M05C

45.0

(kN
)
803

M05M

45.0

855

8.13

322

82

26.9

725

40

21.8

685

51

26.9

725

40

M10C

42.7

751

8.38

387

80

46.5

722

9

24.9

677

42

64.3

751

51

M10M

130

900

9.91

478

92

66.3

804

49

36.3

726

74

101

853

22

M15C

78.7

819

10.2

497

87

77.7

818

1

39.4

731

50

128

889

62

M15M

133

934

11.7

542

91

78.5

886

41

53.8

835

60

160

966

20

M20C

76.5

815

14.0

589

82

90.4

917

18

56.6

842

26

M20M

189

923

13.2

553

93

81.8

888

57

50.3

815

73

(mm)

Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN.

91

Flexure
196

992

3

Concrete compressive strength, fc' (ksi)
1.0

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Damage due to ultimate deformation
Damage due to energy dissipation
y=0.6-45.5e-0.72x (R2=1)

0.9

Damage indices

0.8
0.7

y=0.4+45.5e-0.72x (R2=1)

0.6
0.5
0.4
0.3
0.2
0.1
0.0
40

50

60

70

80

90 100 110 120 130

Concrete compressive strength, fc' [MPa]
Figure 3.35 Concrete strength effect on damage index due to ultimate deformation and
hysteretic energy dissipation.

3.6 Conclusions
Eight cantilever walls were tested with design concrete compressive strengths of 34,
69, 103, and 138 MPa (5, 10, 15 and 20 ksi) under cyclic and monotonic loading to study the
effect of high-strength-concrete (HSC) and load reversals on the inelastic web-crushing
capacity of structural walls.
Two conclusions are offered based on the experimental study presented in this
chapter:
1.

High-strength-concrete can effectively delay web-crushing shear failures in

structural walls thus allowing the system to attain stable inelastic force-displacement
response before failure. This response is possible by the strength gained by the shear92

carrying concrete struts and their anchorage into well-confined boundary elements, which in
turn govern the inelastic flexural response of the system. The result is a stable and
dependable ductile response, which supports the research hypothesis and the possibility of
accepting what can be called „ductile shear failures‟ as acceptable inelastic failure mechanisms
for seismic design.
2.

Second, web crushing at moderate displacement ductilities can be reliably

attained for HSC walls under cyclic loading. Comparison of cyclic and monotonic test results
reveal that cyclic loading significantly curtails the compression capacity of the inclined shearresisting struts in HSC walls. Such effect is attributed to the lower fracture toughness of
HSC, which leads to rapid shear strength and stiffness degradation. Such behavior was most
noticeable for concretes with compressive strength over 103 MPa (15 ksi). This aspect is
essential in establishing dependable limits to the inelastic web crushing capacity of HSC
structural walls.

93

Chapter 4.
4.1

Experimental Studies on HSC Wall Assemblies under Multidirectional Loading

Summary
High-strength-concrete (HSC) offers the potential in optimizing structural design

and reduce material costs due to the possibility to satisfy design requirements with more
slender elements. However, the use of HSC in seismic regions is not well studied and the
corresponding design guidelines are missing. This chapter presents the seismic performance
of HSC wall assemblies within the context of hollow square bridge piers through a largescale experimental investigation. Two 1/4-scale hollow square bridge pier test units featuring
highly-reinforced boundary elements at four corners and thin connecting webs, were
subjected to diagonal and multi-directional cyclic loading with design concrete strengths of
34 and 138 MPa (5 and 20 ksi), respectively. Both test units exhibited stable ductile behavior
until web crushing at moderate ductility levels. The diagonal web crushing performance was
evaluated based on the test observations of the crack pattern and failure mode and the
hysteretic force-displacement behavior. The degradation of shear stiffness and energy
dissipating capacity were quantified by comparing the response of the pier assembly with
that of a single wall in-plane test to evaluate the load path effect on web crushing capacity.
The ductility levels achieved by the two pier test units confirmed that the effect of HSC in
improving web crushing capacity as evaluated in the single wall test program is also true in
wall assemblies. Nonetheless, the pier tests indicated that web crushing capacity is
significantly compromised by concrete damage and strength degradation under multidirectional loading, an effect that is aggravated due to the three-dimensional load demand.

94

4.2 Experimental Investigation
4.2.1

Wall-Assembly Test Units
The three-dimensional inelastic web crushing capacity of structural wall assemblies

was studied within the context of hollow rectangular piers featuring four connected walls
with heavily confined boundary elements. Two pier test units with design concrete
compressive strengths of 34 and 138 MPa (5 and 20 ksi) were constructed and tested at
NEES MAST laboratory at the University of Minnesota [45]. Based on the difference of
loading protocols, the 34 MPa (5 ksi) test unit was named the Diagonal Pier Test (DPT) unit
and the 138 MPa (20 ksi) pier was named the Biaxial Pier Test (BPT) unit. Elevation and
cross section drawings (with reinforcement details) of the DPT test unit are shown in Figure
4.1. The body (cross-section) of the test units was cast such that it was rotated 45 degrees
with respect to the principal directions of the footing and load stub, as shown in Figure
4.1(a). This detail, as well as the overall dimensions of the test units was dictated by
laboratory construction and test setup constraints. The test unit featured highly-reinforced
and well-confined boundary corner elements with thin connecting walls arranged
symmetrically in the form of a hollow square. Confinement to the concrete in the boundary
elements was provided by steel spirals with a spacing of 50 mm (2 in.) within the plastic
hinge length and 76 mm (3 in.) beyond that. Both test units shared the same cross-section
dimensions as well as the longitudinal and confining steel configurations. Other than the
concrete compressive strength the only difference between the test units was the vertical
spacing of the wall transverse steel. The DPT unit had its wall transverse steel uniformly
distributed at 102 mm (4 in.) along the height, while the spacing was 76 mm (3 in.) for the
BPT unit in order to provide greater diagonal tension resistance under shear. A 13 mm (0.5
in.) gap was provided along the cover concrete in the octagonal boundary elements, except
95

along the side connecting to the walls, to prevent premature spalling of the cover concrete.
A construction photograph in Figure 4.2 shows the reinforcement details of one wall in the
BPT pier. An additional change for the BPT unit, was the provision of 6 #3 secondary
longitudinal steel reinforcement bars were added in each wall panel to help control the
excessive cracking. These bars did not continue into the footing or load block to avoid
adding flexural resistance to the section. Anchorage for these bars was achieved by providing
them with 180-deg. Hooks at their ends.

96

(a)

81"
[2057]

12"
[305]

(b)
3"
[76]

24"
[610]

6"
[152]

3"
[76]

#3 spiral @ 2" o.c.
O.D. = 11"

4"
[102]
2#3 @ 4" o.c.

4"
[102]

3"
[76]
6"
[152]
3"
[76]

4 1/2"
[114]

12 #6 bars

2#3 @ 5" o.c.

12"
[305]

4 1/2"
[114]

23"
[584]

4 1/2"
[114]

4"
[102]

1/2" cover
4"
[102]

Figure 4.1 (a) Elevation of the pier test units; (b) Cross section of DPT unit with reinforcement details.
97

24"
[610]

12"
[305]

Figure 4.2 Reinforcement at BPT column base with foam at the base of the boundary
elements and secondary longitudinal steel in the wall.
4.2.2

Materials
The cast of two similar test units with 34 MPa (5 ksi) normal-strength-concrete and

138 MPa (20 ksi) high-strength-concrete allowed comparison of test results in terms of
structural displacement ductility, concrete damage and shear stiffness degradation. Mix
proportions of the concrete for both test units are given in Table 4-1. The concrete
compressive strength development was monitored by a series of cylinder tests for each batch
of concrete. The corresponding concrete strength development curves for both test units are
shown in Figure 4.3. Tension tests were conducted on each bar type used in the pier units.
Properties of the reinforcement are provided in Table 4-2 with parameters corresponding to
Mander‟s uniaxial steel model [42].

98

Table 4-1 Concrete mixtures for pier test units.
BPT

DPT

Cement Type I, kg (lb)

393 (865)

161 (355)

Fly Ash, kg (lb)

–

51 (112)

Silica Fume, kg (lb)

45 (100)

–

Fine Aggregate, kg (lb)

561 (1236)

740 (1629)

Coarse Aggregate, kg(lb)

817 (1800)

795 (1750)

HRWR Type F, liter (oz)

–

1.1 (37)

Retarder Type D, liter (oz)

1.3 (44)

–

Water, kg (lb)

96 (211)

112 (246)

Water Cementitious Ratio

0.22

53

135

20.0

120

17.5

105

15.0

90

12.5

75

10.0

60

7.5

45

5.0

30

Diagonal pier test
Biaxial pier test

15
0
0

20

40

60

80

100

120

2.5

0.0
140

Days

Figure 4.3 Concrete material strength development curves.

99

Concrete compressive strength (ksi)

Concrete compressive strength [MPa]

Item

Table 4-2 Steel properties and modeling parameters for pier test units.

Test Unit

Bar Name

Size

Longitudinal

(ksi)

(ksi)

(ksi)

#6

64.3

100.7

0.0084

1267

Longitudinal

#3

71.8

112.2

0.0061

1400

Spiral

#3

61.5

99.7

0.0053

1150

BPT

Transverse

#3

72.4

114.9

0.0045

1517

DPT

Transverse

#3

64.9

100.6

0.0064

1142

BPT/DPT

Note: 1 MPa = 0.145 ksi.

4.2.3

Test setup

Figure 4.4 shows an overview of BPT setup at the MAST laboratory at the University of
Minnesota. The MAST facility has the unique feature of being able to apply loading to
structural test units by controlling the six degrees of freedom of a stiff loading crossbeam
through eight servo-controlled actuators, two horizontally in each principal direction and
four vertically at the ends of the crossbeam. A flexure/shear aspect ratio of 2.5 was
maintained by controlling the loading such as to create an inflection point 3.05 m (120 in.)
above the top of the footing or 1.37 m (54 in.) down from the top of the load stub where
the crossbeam was attached. The loading components and moment diagram during tests are
shown in Figure 4.5. A moment, the magnitude of which equals the shear force component
times the distance l, which is the vertical length from the desired point of inflection to the
bottom of the loading crossbeam, was applied to keep the inflection point at the desired
location during testing. During the displacement control phase of loading, the feedback
signal from the displacement transducer at the inflection point in the corresponding
direction was used to control the test. All displacement related results and parameters (e.g.,
100

displacement ductility) in the pier tests are with respect to the measured lateral deformations
at the induced inflection point. The active and constrained degrees of freedom at the top of
the pier column during testing are shown in Figure 4.5 are expressed in Equation (4-1). Both
units were tested under a constant axial load of 434 kips, corresponding to

for 34

MPa (5 ksi) concrete.
2

3

(4-1)

Figure 4.4 BPT Test setup at the NEES Multiaxial subassemblage Testing (MAST)
laboratory.
101

'

Fz'
Fx'
Mx'
My'
Fy'

Mx'

My'

x'

T

y'

y

l
I

u'

l

T

v'
I

y'
l = 1372mm (54")
x
45°
x'

Figure 4.5 Loading components and moment diagram.
4.2.4

Instrumentation
The test units were instrumented to measure flexural deformations, shear

deformations, flexure curvatures, and steel reinforcement strains. Curvature was determined
by measuring the vertical deformations at the edges of the boundary elements along the test
unit height. Global flexural deformation were integrated and averaged based on the
curvature values along the height of the pier. Shear deformations were measured by placing
shear measuring panels on two adjacent sides of the test unit featuring two independent
displacement transducers arranged diagonally at each height level. The displacements at the
inflection point were measured with displacement transducers along the corresponding
directions. The flexural deformation contribution from strain penetration, or bond-slip, was
102

calculated from vertical displacement measurements at the section base. This was achieved
by 2 in. displacement transducers sitting on the cantilevers mounted on the footing surface
away from the strain penetration influence. Strain gauges were used to measure strains in the
transverse reinforcement at target locations in the walls. The curvature and shear
deformation instrumentation on the north-east elevation for both test units is shown in
Figure 4.6(a) and (b).
In addition to the conventional instrumentation noted above, the Metris K600
Dynamic Measuring Machine (DMM) was used to measure the spatial deformation of the
north-west wall of the test unit by tracing the movement of the LED target points mounted
on the wall during test, shown in Figure 4.6(b). Also, the South East wall of the test unit was
placed with high-contrast photogrammetry targets so as to evaluate in-plane deformation
fields by digital photogrammetry using photographs taken by DDC camera at the peak of
each loading pattern, as shown in Figure 4.6(b).

103

(a)
(b)

s

s

LCN120

LCW144
LVNW3D2
24"
LVNW3D1
LCW120

s

LCN84

18"
LCW102
LVNW2D2
18"
LCW84

LCN66

LVNW#

LVNW#
LCN#

LCW#
LVSW#

18"
LVNW2D1
LCW66

LCN102

LCN48

s

LCN36
LCN24
LCN12
LCN02

s

18"
LCW48
12"
LCW36
12" LVNW1D2
LCW24
12"
LVNW1D1
LCW12
12"
LCW02

West

LVSW#
LCS#

North

South

Metris Measurement
(NE Wall)

LCN144

East

LCE#

Photogrametry Measurement
(SE Wall)

(a)

104

LCN144

s

Figure 4.6 (a) Curvature and shear deformation instrumentation, North-East elevation; (b) Cross section with instrumentation on
each side.

LCW144
LVNW3D2
24"

4.2.5

Loading Protocol
Both piers were tested quasi-statically according to incrementally increasing, fully

reversed cyclic loading patterns. The 34 MPa (5 ksi) pier, identified as the diagonal pier test
(DPT) unit, was loaded about the section‟s diagonal axis (an axis 45o with respect to wall
principal directions) with the loading sequence shown in Figure 4.7(a). Four initial loading
excursions in load control were applied diagonally along the South-North direction to
,

,

and

, where

is the lateral force to first yield in the diagonal

direction. Loading was then switched to displacement control at the inflection point with the
loading sequence from O→A→B→O, where O represents the origin or zero deformation.
Loading continued by imposing two cycles at each of the ductility level increments of

=

1, 1.5, 2, 3, 4, and 6 or until failure.
The 138 MPa (20 ksi) pier, identified as biaxial pier test (BPT) unit, was loaded with a biaxial
loading protocol with the loading sequence shown in Figure 4.7(b). Four initial load cycles in
load control were applied diagonally along East-West direction to
. Then three cycles at

,

, and

,

,

and

were applied along the longitudinal direction

(SE-NW), the transverse direction (NE-SW) and the other diagonal direction (North-South),
respectively; where

and

are the first yield forces in the principal longitudinal and

transverse directions. The test was switched to displacement control at the inflection point
following the loading sequence described by the incremental letter sequence shown in Figure
4.7(b), that is from O→A→…→H→O. One full bi-axial loading cycle was applied at the
displacement ductility levels of

= 1, 1.5, 2, 3, 4, and 6 until failure of the test unit.

105

5

(a)

East

South
A

4

µ  Longitudinal Direction

3
2
1
0
-1
-2
-3
-4

B

West

North

-5
-5

-4

-3

-2

-1

0

1

2

3

4

5

µ  Transverse Direction
10

(b)

East

South

8
A

µ  Longitudinal Direction

6

C
D

4
2
F

0

E

-2
-4
G

-6

H

B

-8

West

North

-10
10

8

6

4

2

0

-2

-4

-6

-8

-10

µ  Transverse Direction

Figure 4.7 (a) DPT diagonal loading protocol and (b) BPT biaxial loading protocol.
106

The values of first yield forces

and ideal yield displacements

, which are

needed to define the described loading protocols are listed in Table 4-3. A detailed
explanation on the assumptions and methods to determine these quantities can be found in
reference [17].
Table 4-3 Values of force at first yield and ideal yield displacement.

Test Unit

Direction

DPT

kN (kips)

mm (in.)

L

823 (185.0)

13.7 (0.54)

DPT

T

823 (185.0)

13.7 (0.54)

DPT

D

1164 (261.6)

19.6 (0.77)

BPT

L

1384 (310.9)

17.3 (0.68)

BPT

T

1384 (310.9)

17.3 (0.68)

BPT

DL

1028 (231.1)

13.0 (0.51)

BPT

DT

1028 (231.1)

13.0 (0.51)

BPT

D

1454 (326.8)

18.3 (0.72)

4.3 Experimental Results
Both pier test units behaved in a ductile manner before web crushing failure, which
was accompanied by spalling of wall concrete and gradual loss of the load-carrying capacity.
The DPT unit failed on the second excursion to point A at displacement ductility
when the load-carrying capacity dropped 17% in
excursion to point B at

=4

direction and the BPT unit failed on the

= 6 (see Figure 4.7) when the load-carrying capacity dropped

over 20% in both principal directions. Figure 4.8 shows the cracking pattern and the web
crushing failure mode of the DPT unit. The flexural demands on the pier led to concrete
107

spalling on the compression boundary elements and the NE wall, which mainly concentrated
at the bottom of the specimen. The inelastic flexure-shear cracking effects were observed on
all walls and the critical damage and concrete spalling in the SW wall led to the ultimate
failure of the test unit. The gradual loss of the load-carrying capacity demonstrated a ductile
web crushing failure.
Figure 4.9 shows the cracking pattern and the web crushing failure mode of BPT
unit. Web crushing in this unit initiated at the region next to the compression boundary
element, which was severely degraded due to the multi-directional loading demands. The
failure, however, was limited to the plastic hinge region of the walls. Spalling of the wall
cover concrete was observed on the NW side at the interface of the wall and boundary
element. Ultimate failure of the test unit was due to rupture of the transverse spiral
reinforcement immediately above the critical section and subsequent longitudinal bar
buckling on the West compression boundary element when loading toward A at
upon completing a full cycle at

= 8

= 6.

The hysteretic force-displacement response of the DPT and BPT units along their
principal axes is shown in Figure 4.10 and Figure 4.11, respectively. The hysteretic loops of
the BPT unit at

= 4 are isolated and shown in Figure 4.12, which shows the distinct

feature from unloading effects while transitioning from loading points G to H in the
longitudinal direction and D to E in the transverse direction. This phenomenon illustrates
the interactions of the flexural behavior of the test unit under biaxial loading, which leads to
what are considered important effects of unloading and reloading of the longitudinal steel
and the change of the stress state in the compression zone. Table 4-4 and Table 4-5 show
the tabulated force and displacements values at peaks of the hysteretic loops.

108

Representative curvature profiles of the BPT and DPT pier test units in different
loading directions at

= 4 are shown in Figure 4.13. It is interesting to see that the

curvatures profiles along the height of the test units are very close to each other, which
verifies the equivalent nature of the loading protocols and justifies comparing results from
both tests at equal ductility levels.
Figure 4.14 shows a contour plot of the in-plane minimum principal strains, E2, as
well as the test photograph showing the with LED Krypton targets at

= 4 x 1 for the

DPT unit. The targets define the mesh for a displacement field analysis of the wall. The
three-dimensional coordinates obtained from the Krypton measuring system were used to
compute the Green-Lagrange strain within the framework of the finite element method. In
the same way, Figure 4.15 shows the E2 contour at

= 4 x A for the BPT unit. The strain

contours are not only the result of a quite promising non-contact instrumentation strategy,
but provide interesting insight into the behavior of the test unit. It can be noted that the
maximum principal strains for the DPT in Figure 4.14 are indeed very close to the expected
maximum concrete strain of 0.0025, on average, at the compression zone at the bottom of
the wall. The DPT unit failed at this load level but on the SW wall (the Krypton measuring
system was installed in the NE wall). Conversely, the strain contour for the BPT unit in
Figure 4.15 shows that the strains are quite high at the bottom compression zone of the wall
(approximately 0.006). Indeed, attention to the photograph in the same figure shows that the
bottom of the wall had begun to crush due to the biaxial loading demands (compression
demands from flexural excursions on the NE wall). The BPT unit went on to resist a full
cycle of loading at

4 and eventually failed upon a diagonal cycle at

6. Yet, the

noted strain levels are indicative of the type of damage that the biaxial loading pattern has in
curtailing the web crushing capacity of the walls in the pier unit.
109

NW wall

SW wall

SE wall

NE

NW

SE

SW
Figure 4.8 DPT web crushing failure at

110

=4 x 2 loading toward A.

NE wall

NE wall

NW wall

SW wall

SE

NE

SW

NW
Figure 4.9 BPT web

crushing failure at

111

=6 loading toward B.

SE wall

(a)

Drift at the inflection point

Actuator longitudinal force [kN]

1600
1200

Fy

800

Fy'

4x2A

-400
-800
-1200

300
200
100

400
0

400

Ductility
1
-2
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

0
-100

-Fy'

-200

-Fy

4x2B

-300

-1600

Actuator longitudinal force (kips)

-3% h -2% h -1% h 0% h 1% h 2% h 3% h

-400
-100 -80 -60 -40 -20

0

20 40 60 80 100

Longitudinal displacement at the inflection point [mm]

(b)

Drift at the inflection point

Actuator transverse force [kN]

1600
1200

Fy

800

Fy'

4x2B

-400

Ductility

-2 -1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-800
-1200

300
200

400
0

400

100
0
-100

-Fy'
4x2A

-200

-Fy

-300

-1600

Actuator transverse force (kips)

-3% h -2% h -1% h 0% h 1% h 2% h 3% h

-400
-100 -80 -60 -40 -20

0

20 40 60 80 100

Transverse displacement at the inflection point [mm]

Figure 4.10 Hysteretic loops of DPT: (a) longitudinal axis and (b) transverse axis.

112

(a)
-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

1200

4.0

6A

Fy
Fy'

1600

3.0

400
300
200

800
400

Ductility

0

-7 -6 -5 -4 -3 -2 -1 0 1 2
-2
-400

3

4

5

6

7

-800

100
0
-100
-200

-1200

-300

-Fy'
-Fy

6B

-1600
-2000

-400

Actuator longitudinal force (kips)

Actuator longitudinal force [kN]

2000

Drift ratio /L (%)

-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120
Measured longitudinal displacement [mm]

(b)
-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

Fy
Fy'

1600
1200

3.0

4.0
400

6A

300
200

800
400

Ductility

0

-7 -6 -5 -4 -3 -2 -1 0
-400

1 2

3

4

5

6

7

-800

-2000

0
-100
-200

-1200
-1600

100

6B

-Fy'
-Fy

-300

Actuator transverse force (kips)

Actuator transverse force [kN]

2000

Drift ratio /L (%)

-400

-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120
Measured transverse displacement [mm]

Figure 4.11 Hysteretic loops of BPT: (a) longitudinal axis and (b) transverse axis.

113

Drift ratio

L (%)

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
2000
A
Fy
400
1600
C
Fy'
300
1200
200
800
D
100
400
Ductility
F
0
0
-4
-2
-1
0
2
3
4
-3
1
-400
-100
E
-800
-200
-1200
-1600

B

H
G

-2000

-300

-Fy'
-Fy

-75 -60 -45 -30 -15

0

15

-400
30

45

60

Actuator longitudinal force (kips)

Actuator longitudinal force [kN]

(a)

75

Measured longitudinal displacement [mm]
Drift ratio

L (%)

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
2000
F 400
Fy
1600
A
Fy'
300
1200
200
800
G
100
400
Ductility
C
0
0
-4
-3
-2
-1
0
1
2
3
4
-400
-100
H
-800
-200
-1200
-1600
-2000

E

B D

-75 -60 -45 -30 -15

-300

-Fy'
-Fy
0

15

Actuator transverse force (kips)

Actuator transverse force [kN]

(b)

-400
30

45

60

75

Measured transverse displacement [mm]

Figure 4.12 Hysteretic loops of BPT at

=4: (a) longitudinal axis and (b) transverse axis.

114

Table 4-4 DPT unit: peak force-displacement values.
Level

ΔL
(in.)

FL
(kips)

ΔT
(in.)

FT
(kips)

+1/4 FyD′

0.030

46.2

-0.031

-47.1

-1/4 FyD′

-0.026

-46.9

0.015

45.4

+1/2 FyD′

0.087

92.4

-0.087

-92.5

-1/2 FyD′

-0.074

-92.6

0.062

91.5

+3/4 FyD′

0.192

137.0

-0.192

-138.9

-3/4 FyD′

-0.177

-137.8

0.161

139.0

+FyD′

0.342

183.6

-0.355

-185.0

-FyD′

-0.327

-183.6

0.300

185.1

μΔ = +1 x 1

0.518

224.2

-0.534

-223.8

μΔ = -1 x 1

-0.557

-231.4

0.521

233.5

μΔ = +1 x 2

0.516

215.5

-0.535

-215.8

μΔ = -1 x 2

-0.557

-221.3

0.518

224.3

μΔ = +1.5 x 1

0.783

257.6

-0.805

-257.0

μΔ = -1.5 x 1

-0.834

-261.5

0.792

265.7

μΔ = +1.5 x 2

0.788

246.4

-0.810

-246.1

μΔ = -1.5 x 2

-0.838

-252.0

0.793

256.0

μΔ = +2 x 1

1.063

278.5

-1.076

-277.7

μΔ = -2 x 1

-1.113

-279.5

1.069

284.9

μΔ = +2 x 2

1.063

262.3

-1.078

-262.2

μΔ = -2 x 2

-1.111

-266.4

1.069

272.2

μΔ = +3 x 1

1.622

302.6

-1.613

-303.4

μΔ = -3 x 1

-1.657

-300.8

1.627

306.5

μΔ = +3 x 2

1.629

277.5

-1.614

-279.5

115

Table 4-4 (cont‟d).
μΔ = -3 x 2

-1.661

-283.7

1.627

290.5

μΔ = +4 x 1

2.189

307.7

-2.133

-310.1

μΔ = -4 x 1

-2.199

-308.8

2.161

316.1

μΔ = +4 x 2

2.213

272.3

-2.121

-281.0

μΔ = -4 x 2

-2.295

-233.3

2.066

271.9

μΔ = +6 x 1

3.591

160.8

-2.909

-336.5

μΔ = -6 x 1

-3.606

-113.0

2.836

307.1

μΔ = +6 x 2

3.569

106.8

-3.067

-291.9

μΔ = -6 x 2

-3.412

-95.3

3.000

174.3

116

Table 4-5 BPT unit: peak force-displacement values.

+1/4 FyD′

ΔL
(in.)
0.026

FL
(kips)
57.5

ΔT
(in.)
0.024

FT
(kips)
58.6

-1/4 FyD′

-0.033

-58.2

-0.026

-56.7

+1/2 FyD′

0.081

115.4

0.081

115.7

-1/2 FyD′

-0.090

-115.7

-0.083

-114.4

+3/4 FyD′

0.181

172.5

0.181

173.1

-3/4 FyD′

-0.191

-173.3

-0.185

-172.2

+FyD′

0.333

230.4

0.329

231.1

-FyD′

-0.338

-229.9

-0.336

-231.0

+FyL′

0.521

311.2

-0.026

-12.1

+FyL′

-0.543

-311.0

-0.016

19.1

+FyT′

-0.059

-25.0

0.528

310.8

+FyT′

-0.057

-7.9

-0.529

-310.8

μΔ = A x 1

0.495

266.3

0.492

266.7

μΔ = B x 1

-0.505

-260.8

-0.503

-263.8

μΔ = C x 1

0.682

338.7

-0.023

-7.7

μΔ = D x 1

0.683

290.4

-0.328

-203.7

μΔ = E x 1

0.328

96.6

-0.681

-292.8

μΔ = F x 1

0.003

-18.9

0.668

336.2

μΔ = G x 1

-0.497

-259.0

0.534

228.1

μΔ = H x 1

-0.680

-284.2

0.029

-53.8

μΔ = A x 1.5

0.743

304.7

0.746

291.4

μΔ = B x 1.5

-0.758

-289.4

-0.755

-298.8

μΔ = C x 1.5

1.025

362.1

-0.020

-0.6

μΔ = D x 1.5

0.797

227.7

-0.770

-307.4

μΔ = E x 1.5

0.027

-82.6

-1.033

-299.4

μΔ = F x 1.5

0.001

-10.7

1.006

359.3

μΔ = G x 1.5

-0.761

-294.6

0.795

230.2

μΔ = H x 1.5

-1.015

-288.6

0.036

-83.3

μΔ = A x 2

1.002

331.7

1.007

297.0

Level

117

Table 4-5 (cont‟d).
μΔ = B x 2

-1.011

-296.3

-1.014

-318.6

μΔ = C x 2

1.363

371.9

-0.022

-0.7

μΔ = D x 2

1.072

222.5

-1.034

-330.7

μΔ = E x 2

0.034

-106.1

-1.334

-281.4

μΔ = F x 2

-0.002

-7.7

1.334

368.9

μΔ = G x 2

-1.006

-316.4

1.053

223.0

μΔ = H x 2

-1.355

-290.0

0.045

-105.3

μΔ = A x 3

1.513

364.8

1.508

332.5

μΔ = B x 3

-1.511

-329.2

-1.536

-344.8

μΔ = C x 3

2.043

389.4

-0.011

1.6

μΔ = D x 3

1.586

208.4

-1.531

-359.3

μΔ = E x 3

0.041

-135.8

-2.028

-277.4

μΔ = F x 3

-0.014

-4.6

2.031

390.6

μΔ = G x 3

-1.525

-350.0

1.600

209.8

μΔ = H x 3

-2.042

-274.7

0.058

-140.4

μΔ = A x 4

2.038

373.2

1.999

329.6

μΔ = B x 4

-2.003

-325.8

-2.041

-350.0

μΔ = C x 4

2.727

396.0

0.008

-0.9

μΔ = D x 4

2.004

173.9

-1.922

-360.0

μΔ = E x 4

0.051

-141.9

-2.710

-287.9

μΔ = F x 4

-0.015

1.8

2.725

398.0

μΔ = G x 4

-2.035

-358.9

2.127

189.2

μΔ = H x 4

-2.711

-256.5

0.058

-159.5

μΔ = A x 6

3.043

385.5

2.962

350.5

μΔ = B x 6

-2.979

-292.2

-3.073

-307.9

μΔ = C x 6

4.082

353.6

0.023

-15.9

μΔ = D x 6

3.454

130.2

-2.650

-272.3

μΔ = E x 6

0.121

-142.5

-3.977

-216.0

μΔ = F x 6

0.043

-2.5

4.088

277.5

μΔ = G x 6

-2.786

-235.5

3.119

68.5

μΔ = H x 6

-4.030

-197.4

0.181

-122.2

118

Averaged curvature (x 10-3 rad/in.)
0.1

0.2

0.3

0.4

0.5

3000

Height above footing [mm]

2700
2400
2100
BPT
y|=9.612e-5 rad/in.

1800
1500
1200
900
600
300
0
0

5

10

15

20

0.6

0.7

0.8

0.9

120
110
BPT longitudinal
100
BPT transverse
90
BPT diagonal
BPT sweeping diag. 80
DPT diagonal
70
60
50
40
30
20
10
0
25
30
35

Height above footing (in.)

0.0

Average curvature [x10-6 rad/mm]

Figure 4.13 Curvature profiles of BPT and DPT in different directions.

119

2000
1000

-2000
-6000

600
-10000

400

-14000

200
0
-400

E2

Wall Height (mm)

800

-18000
-200

0

200

400

Wall Width (mm)
Figure 4.14 DPT in-plane minimum principal strain, E2 at

120

=4x1.

1200

0

800

-4000

600

-6000

400

-8000

E2

-2000

Wall Height (mm)

1000

200
-400

-10000
-200

0

200

400

Wall Width (mm)
Figure 4.15 BPT in-plane minimum principal strain, E2 at

=4xA.

4.4 Discussion
Figure 4.16 compares the force-displacement envelopes of the BPT pier unit along
different directions. It is interesting to note that the behavior of the test unit in the sweeping
diagonal direction is close to that of the principal directions instead of the straight diagonal
directions. The coupling between the two orthogonal directions decreases the stiffness in the
sweeping diagonal direction. By comparison, loading along the other directions passes
through the centroid of the cross section and no coupling exists. Figure 4.17 compares the
121

DPT hysteretic loop with the BPT envelopes in the diagonal and sweeping diagonal
directions. It can be seen that the BPT has higher load-carrying capacity than the DPT in the
diagonal direction. In the sweeping diagonal direction, the BPT experienced complicated
unloading behavior due to the eccentricity of the loading.
The shear stiffness degradation and the energy dissipating capacity of the inelastic
shear mechanism of the test units under various loading patterns are of interest in this study.
The displacement ductility limit of the hollow piers before web crushing failure is directly
related to those two factors. Figure 4.18 and Figure 4.19 show the hysteretic loops of both
test units for flexure and shear deformations, respectively. It can be seen that flexure (Figure
4.18(a) and Figure 4.19(a)) dominates the force-displacement behavior of both test units,
while comparison of Figure 4.18(b) and Figure 4.19(b) indicates that the DPT unit exhibited
larger inelastic shear deformations and energy dissipating capacity than the BPT unit.
To study the shear stiffness degradation of the DPT, a linear regression analysis was
conducted on the unloading curves of the shear hysteretic loops. The unloading stiffness
degradation of the DPT against displacement ductility is shown in Figure 4.20(a). It is
noteworthy that the damage degradation from the first loading peak (Point A) with respect
to the second loading peak (Point B) progressively decreases, indicating overall damage to
the structure. For the BPT unit, the “butterfly” loading pattern disturbed the unloading
scheme of the test unit and the unloading stiffness is difficult to evaluate. Therefore, the
degradation of the secant stiffness was studied since plastic shear deformations were small.
The secant stiffness degradation of the BPT test unit is shown in (b). Of interest here is the
fact that the degradation follows a quadratic trend, compared to the linear degradation that
was seen in structural walls subjected to in-plane shear demands [16][45]. More discussion
on this observation follows.
122

Hines et al. [17] tested a single cantilever wall (Unit 3A) with the same cross section,
steel reinforcement, and aspect ratio as one of the sides of the pier units in this program.
Figure 4.21 shows the global and the shear hysteretic loops of Unit 3A. A comparison of the
shear hysteretic loops indicates that Unit 3A (scaled by a factor of 2 to account for the
number of walls in the pier units) exhibited less shear stiffness degradation and better
inelastic shear behavior than the piers, which suffered much greater and rapid shear stiffness
degradation due to the mixed flexure-shear cracking under multi-directional loading. The
shear dissipating energy of both piers along their principal directions is compared with that
of Unit 3A in Figure 4.22. It can be seen that BPT had less shear energy dissipating capacity
than the DPT and that both pier tests dissipated much less energy compared to twice the
single wall test. The linear energy dissipation of the in-plane loaded wall compared to the
quadratic degradation of the piers can also be noted.

123

Drift ratio D/L (%)
-4.0

-3.0

-1.0

0.0

1.0

2.0

3.0

4.0
500
400
300
200
100
0
-100
-200
-300
-400
-500

Longitudinal
Transverse
Diagonal
Sweeping diag.

2000
1500

Actuator force [kN]

-2.0

1000
500
0
-500
-1000
-1500
-2000
-2500

Actuator force (kips)

2500

-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120
Measured displacement [mm]

Figure 4.16 Comparison of the BPT force-displacement envelopes in different directions.
Drift ratio D/L (%)

2000
Actuator force [kN]

1500

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

DPT Diagonal
BPT Diagonal
BPT Sweeping Diag.

1000
500
0
-500
-1000
-1500
-2000
-2500

500
400
300
200
100
0
-100
-200
-300
-400
-500

Actuator force (kips)

2500

-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120
Measured displacement [mm]

Figure 4.17 DPT hysteretic loop compared to BPT envelopes in diagonal and sweeping
diagonal direction.

124

Drift at the inflection point
-1% h

0% h

1% h

2% h

400

Actuator longitudinal force [kN]

1600
1200

Fy

300

800

Fy'

200

400

100

Ductility

0

-4

-400

-3

-2

-1

1

0

2

3

4

0
-100

-800

-Fy'

-200

-1200

-Fy

-300

-1600

Actuator longitudinal force (kips)

-2% h

-400
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Longitudinal flexural displacement at the inflection point [mm]

Longitudinal shear displacement at the inflection point (in.)
-0.5

0.0

0.5

1.0

Actuator longitudinal force [kN]

1600

400

1200

Fy

300

800

Fy'

200
100

400
0

0

-400

-100

-800

-Fy'

-200

-1200

-Fy

-300

-1600

-400
-25 -20 -15 -10

-5

0

5

10

15

20

25

Longitudinal shear displacement at the inflection point [mm]

Figure 4.18 Hysteretic loops of DPT in (a) flexure and (b) shear.

125

Actuator longitudinal force (kips)

-1.0

Drift at the inflection point
-1% h

0% h

1% h

2% h
400

Fy
Fy'

1600
1200

300
200

800
400

100

Ductility

0

-3

-400

-2

-1
-1

1

0

2

3

0
-100

-800

-200

-1200

-300

-Fy'
-Fy'

-1600
-2000

Actuator transverse force (kips)

Actuator transverse force [kN]

-2% h
2000

-400

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60
Transverse flexural displacement at the inflection point [mm]

Transverse shear displacement at the inflection point (in.)
-0.5

0.0

0.5

1.0

Fy
Fy'

1600
1200

400
300
200

800
400

Ductility

0

100
0

-400

-100

-800

-200

-1200

-300

-Fy'
-Fy

-1600
-2000
-25 -20 -15 -10

-5

0

5

-400
10

15

20

25

Transverse shear displacement at the inflection point [mm]

Figure 4.19 Hysteretic loops of BPT in (a) flexure and (b) shear.

126

Actuator transverse force (kips)

Actuator transverse orce [kN]

-1.0
2000

350000

2000
1800

300000

1600

250000

1400
1200

200000

1000
150000

800

100000

600

Shear unloading stiffness (kips/in)

Shear unloading stiffness [kN/m]

2200
Unloading after A
Unloading after B

400
0

1

2

3

4

5

Displacement ductility
3200

500000

2800

450000

2400

400000

2000

350000
300000

1600

250000
200000

1200

150000

800

100000

Shear secant stiffness (kips/in)

Shear secant stiffness [kN/m]

550000

400
0

1

2

3

4

5

Displacement ductility

Figure 4.20 Multidirectional loading effects: (a) Shear unloading stiffness degradation of
DPT in diagonal direction, and (b) Shear secant stiffness degradation of BPT in transverse
direction.

127

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
800

Fy
Fy'

600

200
150
100

200

50
Ductility

0

-4

-200

-2
-2

-3

-1
-1

1

0

2

3

4

-50

-400

-100

-Fy'
-Fy

-600
-800

0

Force (kips)

Force [kN]

400

-150
-200

-100 -80 -60 -40 -20

0

20 40 60 80 100

Displacement [mm]

Shear displacement (in.)
-1.5

-1.0

-0.5

800

0.0

0.5

1.0

1.5

Fy
Fy'

600

200
150
100

200

50

0

0

-200

-50

-400

-100

-Fy'
-Fy

-600
-800

Force (kips)

Force [kN]

400

-150
-200

-30

-20

-10

0

10

20

30

Shear displacement [mm]

Figure 4.21 Hysteretic loops of Unit 3A (a) Global behavior (b) Shear behavior (Hines 2004).

128

Shear dissipating energy [kN-m]

3A (x2)
DPT
BPT

80
70

700
600

60

500

50

400

40

300

30
20

200

10

100

0

Shear dissipating energy (kips-in)

800

90

0
0

1

2

3

4

5

Displacement ductility

Figure 4.22 Comparison of shear dissipating energy of pier tests and single wall test.

4.5 Conclusions
The effect of multi-directional loading on the inelastic web crushing capacity of HSC
wall assemblies was experimentally investigated within the context of hollow square bridge
piers through tests on two 1/4-scale units subjected to diagonal and multi-axial cyclic
loading. Results have confirmed some of the advantages hypothesized for using HSC but
have also brought forth important issues in establishing performance limits to inelastic web
crushing in walls under three-dimensional shear demands. The following conclusions are
offered based on the noted results and findings to date.
1. Both pier test units exhibited a stable ductile response up to moderate ductility levels
before experiencing a web crushing shear failure. The three dimensional force-resisting
mechanism provided by the integrated response of the boundary elements to resist flexure
and the walls to resist shear was efficient and can be relied upon for new designs.

129

2. The effect of high-strength-concrete (HSC) in delaying the web-crushing shear
failure in the BPT pier compared to the DPT unit was verified even though the BPT wall
assembly was loaded under more severe multi-directional demands.
3. The BPT unit exhibited less inelastic shear deformation and more damage due to the
application of HSC and the multi-directional loading pattern compared to the DPT unit.
Therefore, the use of HSC to increase the ductility level was impaired by the severe biaxial
demands and the higher damage susceptibility of HSC.
4. Compared to a prior single wall test by Hines et al. [17], both pier test units exhibited
rapid shear stiffness degradation, which is attributed to the damage accumulation under
multi-directional loading and the lower fracture toughness of HSC.

130

Chapter 5.
5.1

Nonlinear Finite Element Modeling of HSC Structural
Walls

Summary
Nonlinear finite element modeling (NLFEM) on the web crushing capacity of high-

strength-concrete (HSC) structural walls was conducted and described in this chapter. Two
parallel approaches were followed for this task. One is three-dimensional continuum-type
finite element modeling with ABAQUS using a concrete damaged plasticity (CDP) model
for concrete. The other is in-plane modeling with VecTor2 based on the Modified
Compression Field Theory (MCFT). The analyses are described from the aspects of
geometrical modeling, material properties, solution controls, and results. The results are
compared with test data at the global and local levels to evaluate their performance in
predicting flexural and shear behavior of structural walls failing in web crushing. Capabilities
and limitations of the methods in evaluating the web crushing failure mechanism and
capacity of structures were investigated. The continuum-type plasticity-based 3D models in
ABAQUS were able to capture the global behavior of the single walls under monotonic
loading well. However, issues related to the numerical technique of viscoplastic
regularization and the behavior of the reinforced concrete structures under cyclic loading
remain. The need for modifications to the concrete damaged plasticity model in ABAQUS in
order to capture the inelastic shear behavior is recommended. On the other hand, VecTor2
analyses could model the behavior of the structural walls under both monotonic and cyclic
loading well. Even the inelastic shear behavior in the plastic hinge region was captured.
However, the concrete strength degradation under cyclic loading and the web crushing
capacity are not considered appropriately and the web crushing capacity of the walls under
cyclic loading could not be predicted. Both analytical methods revealed that the concrete
131

stress at web crushing was only a small portion of the compressive strength of HSC due to
the complicated stress state in the web crushing region. The analytical results confirm that
web crushing in the form of an inelastic shear failure is caused by the flexure-shear
interaction effects induced in the web of the structural walls.

5.2 Introduction
Finite element modeling (FEM) is commonly conducted in order to improve the
understanding of observed experimental phenomena or predict the complicated behavior of
designed structures. With the rapid growth of computing resources and the use of parallelcomputing, the analysis of large-scale reinforced concrete structures is becoming realistic and
cost-effective.
The earliest application of FEM to reinforced concrete structures was by Ngo and
Scordelis in 1967 [46]. Intensive research has been carried out since the 1970s on the special
features of finite element analysis of reinforced concrete structures, including: constitutive
relationships, failure theories, multi-axial stress theories, modeling of reinforcement,
behavior on the interface between reinforcement and concrete, crack representation,
mechanism of shear transfer, cyclic and dynamic loading effects, and the time-dependent
effects of creep, shrinkage, and temperature variation.
However, the application of FEM on even traditional reinforced concrete structures
has some problems due to the complicated nature of concrete material with crack
propagation, slip across cracks, localization of stresses, and blond-slip behavior of the
reinforcement. The weakness of concrete in tension leads to early cracking and the
homogenous continuum assumption in conventional finite element theory is not valid in a
strict sense. Moreover, to study the seismic behavior of the structures, it is essential for the

132

models to be able to consider the effect of cyclic loading on the degrading concrete, which is
still a challenge for most analysis models and procedures.
The earliest finite element analyses of reinforced concrete structures introduced the
concept of discrete cracks [46]. However, due to the difficulty in updating the mesh with the
progress of cracking with realistic computer resources, the application and development of
the discrete crack approach is very limited. On the other hand, the smeared crack concept
was introduced [47], which is compatible with the framework of the FEM since the mesh
topology does not need to be updated during the analysis. Within the framework of smeared
crack models, two concepts of cracking have been introduced: the fixed crack and the
rotating crack concepts. The fundamental difference between them lies in the orientation of
the cracks, which is kept constant in the fixed crack model and it is continuously updated in
the rotating crack model. Another difference is related to the consideration of shear effects.
In the fixed crack model, shear on the crack surface makes the axes of principal strain and
stress non-coincident. This problem is solved by introducing a unique shear term that
enforces coaxiality between principal stresses and strains. Nonetheless, use of the smeared
rotating crack model is gaining popularity as indicated by the prevalence of the modified
compression field theory (MCFT), which will be discussed later.
In the 3D modeling of reinforced concrete structures, a lot of effort has been
dedicated in developing constitutive relationships for concrete materials within the
framework of plasticity, fracture and damage mechanics. A noteworthy contribution was
done by Chen et al. [48] who developed a plastic theory for concrete materials. Generally,
3D modeling has a very high demand on computer resources. Nowadays, the use of high
performance parallel computing is common for the nonlinear analysis of large models.
Unlike the modeling of plain concrete or very-light reinforced concrete structures like dams,
133

the steel reinforcement in structures like bridge girders or structural walls changes the
mechanical properties of the concrete material, which may place the efficiency of 3D
concrete model in doubt. For example, the amount of reinforcement can affect the dowel
effect on concrete and the tension behavior of concrete. Moreover, the confinement
provided by stirrups improves the compressive strength of concrete. The bond and slip
behavior of the embedded reinforcement is also paramount in adequately capturing localized
stress effects as well as the distribution of inelastic actions. Unfortunately, these effects are
very difficult to capture within a strict theoretical mechanics framework and most of the
currently available finite element formulations fail to consider these effects.
On the other hand, phenomoenological models based on moment-curvature analyses
[19] or relatively complicated beam-column-element [49] can work quite well for predicting
the behavior of structures. They do this by capturing macro-scale behavior through
experimentally calibrated models. These models have become increasingly valuable due to
their ability to reliably predict global response. Further, in many cases, complicated 3D
modeling is usually not needed and may not be able to give more accuracy compared to the
much simpler, yet not strictly theoretically justified, phenomenological models. However, a
common difficulty in analyzing reinforced concrete structural walls with finite element
methods is the challenge of capturing inelastic shear behavior. Unlike the response of beams
or slender walls, shear deformations contribute significantly, or dominate, the behavior and
failure in squat or relatively short walls (aspect rations below 3).
In this chapter, two numerical investigations with different FE programs were
conducted in parallel to analyze and model cantilevered structural walls subjected to
flexure/shear effects. The case studies consist of the single wall test units evaluated
experimentally in the first phase of this study (see Chapter 3). One study was conducted
134

using 3D continuum-type plasticity-based FE modeling; the other consisted in performing
2D plane-stress analysis phenomenological-based FE models. In both cases, large general
commercial nonlinear finite element software was used. The program ABAQUS [35] was
used for the 3D continuum plasticity-based FE analyses, with particular use of its concrete
damaged plasticity (CDP) model. The program VecTor2 [39] was used for the 2D planestress phenomenological-based FE analysis, with particular use of its implementation of the
MCFT [28][39]. The analyses in this chapter are described from the aspects of geometrical
modeling, material properties, solution controls, and results. Results were compared with test
data at the global and local levels to evaluate performance of the analytical methods in
predicting flexural/shear behavior of structural walls failing in web crushing. Capabilities and
limitations of the methods in evaluating the web crushing failure mechanism and capacity in
reinforced concrete walls were investigated. Further, the analytical results provided deep
insights and promoted the understanding of web crushing behavior and the inelastic shear
failure mechanism in structural walls with both qualitative and quantitative information.

5.3 ABAQUS Analysis of Single Walls under Monotonic Loading
5.3.1

Geometrical Modeling
Models for the single wall test units were created through ABAQUS CAE. ABAQUS

version 6.9-2 Standard [35] was used to implement all the nonlinear static analyses. Half of
the walls were modeled in view of the symmetry along the central vertical plane of the walls.
An overview of the model is shown in Figure 5.1(a). Two parts were created, one for the
concrete material which included the footing, the column body and the loading block, and
the other for the reinforcement where only the column body steel was explicitly modeled.
Figure 5.1(b) shows the reinforcement part while the concrete part is as shown Figure 5.1(a).
135

Each part was meshed separately with the reinforcement being embedded in the concrete
solid elements. The meshed model is shown in Figure 5.1(c). Both footing and load block
were assigned elastic material properties without reinforcement. The concrete damaged
plasticity (CDP) material model in ABAQUS was used for the column body concrete. The
element type for the concrete parts was a three-dimensional continuum-type solid element
with 8 nodes and reduced integration (C3D8R). The steel reinforcement was modeled with
three-dimensional two-node truss elements with full integration.

136

Figure 5.1 Overview of 3D ABAQUS model: (a) Model geometry; (b) Reinforcement part; (c) Mesh of the concrete part

137

5.3.2

Material Properties
With the use of the CDP model, it is critical to appropriately define the compression

softening and tension stiffening behavior of the concrete material. Very high material
nonlinearity is expected in reinforced concrete walls failing by web crushing, which normally
causes numerical problems and convergence difficulties in ABAQUS Standard. It is almost
essential to introduce some viscoplastic damping to achieve convergence; however, the
viscosity parameter should be as small as possible so that adequate accuracy of the results
can be achieved. Therefore, a trial-and-error on the value of the parameter is usually needed.
The elastic property of the concrete material is defined by the elastic modulus and
the Poisson‟s ratio. To define the CDP model, curves for concrete compression hardening,
compression damage, concrete tension stiffening, tension damage and parameters to
describing the shape of the flow potential and yield function need to be defined. These
parameters include the dilation angle

 the flow potential eccentricity , the ratio of initial

equibiaxial compressive yield stress to initial uniaxial compressive yield stress

, and

the ratio of the second stress invariant on the tensile meridian to that on the compressive
meridian

. The experimental values for tensile and compressive strengths of concrete were

used to generate uniaxial stress-strain curves with the model suggested by Collins et al. for
compression [50] and the exponential model for tension [51]. Concrete damage was defined
by following the recommendations by Lee and Fenves [37].
The steel reinforcement properties were defined based on experimental data on
tension tests of the different bar sizes fitted with uniaxial models. The calibrated parameters
were then used to define the input curves for analysis.

138

The definition of the CDP model is explained in the following. The uniaxial
compressive stress-strain constitutive relationship follows the model proposed by Collins et
al. [50] with the equations shown below.
(5-1)

⁄
where

(5-2)

(psi units)

(5-3)

(psi units)

(5-4)

√

(psi units)

(5-5)

An exponential law is used to model the tension stiffening of concrete material [51].
The corresponding exponential equation is
(

)

(5-6)
(5-7)

where

is the fracture energy of concrete, which is the energy required to propagate a

tensile crack of unit area. The values of

for different strengths of concrete were obtained

from Table 2.1.4, which is provided by the CEB-FIP Model Code 1990 [52]. Table 5-1
shows the

values corresponding to the maximum aggregate size of 8 mm (0.315 in.). A

linear fit equation of the data in Table 5-1 was used to calculate the

values for the

corresponding concrete compressive strengths in the high-strength-concrete wall test units.

139

Table 5-1 Fracture energy of concrete materials
20

(Nm/m2)

28

38

48

58

68

78

88

40

(ksi)

50

65

70

85

95

105

115

Another important part of the CDP model definition are the damage properties.
Cervera et al. [53] proposed a damage model for concrete dam analysis. However, in this
study, the damage evolution rules by Lee and Fenves [37] are referred for the definition of
damage curves. The explanation of the definition is shown in the following.
For damage in both tension and compression

{

},
⁄

0.

/.

√

(5-8)

/1

(5-9)
(5-10)

∫

(5-11)

∫
(

̃ )

(

)

(
0

̃ )

√

(5-12a)

1

(5-12b)

where ̃ is the damage value at the maximum compressive stress and ̃ is corresponding
to

⁄ ,

can be calculated from √

140

and

can be calculated from

√ ( )
and

√

is corresponding to

, where

is corresponding to the maximum compressive stress
⁄ .

Table 5-2 to Table 5-5 show the parameters of the CDP model for concrete and the
steel nonlinear parameters for the four single walls tested under monotonic loading.

141

Table 5-2 The concrete and steel material parameters for M05M.
The parameters of CDP model
28°

0.1
1.16
0.667

0.03

Concrete compression damage
DamageC
Crushing strain
0
0
0.05
0.00022
0.15
0.00068
0.43
0.00239
0.59
0.00378
0.74
0.00554
0.89
0.00740
0.89
0.00792
Concrete tension damage
DamageT
Cracking strain
0
0
0.50
0.00051
0.89
0.00159
0.97
0.00260
0.97
0.00500

M05M
Concrete elasticity
E(ksi)
4006.7
0.2

Concrete compression hardening
Stress (ksi)
Crushing strain
4.00
0
5.12
0.00022
5.65
0.00068
3.45
0.00239
1.88
0.00378
0.83
0.00554
0.39
0.00740
0.32
0.00792
Concrete tension stiffening
Stress (ksi)
Cracking strain
0.515
0
0.258
0.00051
0.052
0.00159
0.011
0.00260
0.011
0.00500

Reinforcing Steel Properties

#8 Steel
E(ksi)

Stress (ksi)
76.32
88.81
97.44
106.14
114.93
123.79
132.74
141.77
150.88
160.07

29000
0.3
Plastic strain
0
0.00689
0.01153
0.01614
0.02073
0.02529
0.02982
0.03433
0.03881
0.04327

#7 Steel
E(ksi)
29000
0.3

Stress (ksi)
Plastic strain
65.05
0
86.68
0.01681
93.65
0.02633
98.11
0.03584
100.92
0.04531
102.75
0.05473
104.06
0.06407
105.12
0.07334
106.11
0.08252
107.09
0.09162

142

#3 Steel
E(ksi)
29000
0.3

Stress (ksi)
Plastic strain
66.63
0
66.97
0.00516
80.22
0.01704
88.52
0.02651
95.09
0.03594
100.19
0.04534
104.07
0.05468
106.98
0.06397
109.14
0.0732
110.77
0.08236
112.05
0.09145

Table 5-3 The concrete and steel material parameters for M10M.
The parameters of CDP model
34.6°
M10M

0.1
Concrete elasticity
1.16
E(ksi)
5472.1
0.667

0.2
0.02

Concrete compression hardening
Concrete compression damage
Stress (ksi)
Crushing strain
DamageC
Crushing strain
9.25
0
0
0
11.58
0.00013
0.048
0.00013
12.5
0.00048
0.16
0.00048
7.21
0.00193
0.509
0.00193
2.26
0.00334
0.773
0.00334
0.19
0.00472
0.929
0.00472
0.19
0.00750
0.929
0.00750
Concrete tension stiffening
Concrete tension damage
Stress (ksi)
Cracking strain
DamageT
Cracking strain
0.804
0
0
0
0.573
0.0003
0.292
0.0003
0.402
0.00059
0.500
0.00059
0.197
0.00116
0.744
0.00116
0.068
0.00199
0.904
0.00199
0.068
0.00320
0.977
0.00320
0.068
0.00500
0.977
0.00500
Reinforcing Steel Properties
#8 Steel
#7 Steel
#3 Steel
E(ksi)
29000
E(ksi)
29000
E(ksi)
29000
0.3
0.3
0.3
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
68.16
0
65
0
68.76
0
68.68
0.008
65.5
0.008
68.99
0.003
79.43
0.017
75.84
0.017
74.86
0.007
88.21
0.027
83.64
0.027
86.57
0.017
95.22
0.036
89.37
0.036
95.82
0.026
100.65
0.045
93.44
0.046
102.95
0.036
104.72
0.055
96.26
0.055
108.26
0.045
107.66
0.064
98.18
0.064
112.11
0.054
109.7
0.073
99.53
0.074
114.8
0.064
111.12
0.082
100.57
0.083
116.67
0.073
112.2
0.091
101.51
0.092
118.01
0.082
119.13
0.091

143

Table 5-4 The concrete and steel material parameters for M15M.
The parameters of CDP model
25°
M15M

0.1
Concrete elasticity
1.16
E(ksi)
6075.4
0.667

0.02


Concrete compression hardening
Concrete compression damage
Stress (ksi)
Crushing strain
DamageC
Crushing strain
11.99
0
0
0
14.66
8.60E-05
0.026
8.60E-05
16.10
0.00042
0.130
0.00042
13.81
0.00098
0.273
0.00098
8.15
0.00216
0.530
0.00216
3.51
0.00317
0.734
0.00317
0.19
0.00447
0.934
0.00447
0.19
0.00600
0.934
0.00600
Concrete tension stiffening
Concrete tension damage
Stress (ksi)
Cracking strain
DamageT
Cracking strain
0.898
0
0
0
0.664
0.00029
0.265
0.00029
0.449
0.00065
0.500
0.00065
0.255
0.00116
0.707
0.00116
0.158
0.00157
0.812
0.00157
0.077
0.00219
0.903
0.00219
0.077
0.00300
0.903
0.00300
0.077
0.00380
0.903
0.00380
0.077
0.00500
0.903
0.00500
Reinforcing Steel Properties
#8 Steel
#7 Steel
#3 Steel
E(ksi)
29000
E(ksi)
29000
E(ksi)
29000
0.3
0.3
0.3
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
85.81
0
61.31
0
69.95
0
93.48
0.017
61.78
0.008
70.2
0.004
98.73
0.026
73.45
0.017
76.38
0.007
103.95
0.036
81.97
0.027
89.17
0.017
109.22
0.045
88.18
0.036
98.66
0.026
114.55
0.054
92.57
0.046
105.51
0.036
119.96
0.064
95.57
0.055
110.28
0.045
125.45
0.073
97.59
0.064
113.53
0.054
131.03
0.082
98.98
0.074
115.72
0.064
136.68
0.091
100.03
0.083
117.26
0.073
100.96
0.092
118.47
0.082
119.56
0.091
144

Table 5-5 The concrete and steel material parameters for M20M.
The parameters of CDP model
25°
M20M

0.1
Concrete elasticity
1.16
E(ksi)
0.667
6075.4

0.2
0.02

Concrete compression hardening
Concrete compression damage
Stress (ksi)
Crushing strain
DamageC
Crushing strain
11.99
0
0
0
14.66
8.60E-05
0.026
8.60E-05
16.10
0.00042
0.130
0.00042
13.81
0.00098
0.273
0.00098
8.15
0.00216
0.530
0.00216
3.51
0.00317
0.734
0.00317
0.19
0.00447
0.934
0.00447
0.19
0.00600
0.934
0.00600
Concrete tension stiffening
Concrete tension damage
Stress (ksi)
Cracking strain
DamageT
Cracking strain
0.898
0
0
0
0.664
0.00029
0.265
0.00029
0.449
0.00065
0.500
0.00065
0.255
0.00116
0.707
0.00116
0.158
0.00157
0.812
0.00157
0.077
0.00219
0.903
0.00219
0.077
0.00300
0.903
0.00300
0.077
0.00380
0.903
0.00380
0.077
0.00500
0.903
0.00500
Reinforcing Steel Properties
#8 Steel
#7 Steel
#3 Steel
E(ksi)
29000
E(ksi)
29000
E(ksi)
29000
0.3
0.3
0.3
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
65.57
0
64.86
0
63.64
0
65.78
0.003
65.1
0.004
63.77
0.002
71.98
0.007
70.36
0.008
71.07
0.007
83.26
0.017
81.54
0.017
81.82
0.017
91.89
0.026
90.25
0.026
90.36
0.026
98.3
0.036
96.84
0.036
96.95
0.036
102.93
0.045
101.67
0.045
101.91
0.045
106.17
0.055
105.11
0.055
105.5
0.055
108.39
0.064
107.49
0.064
108.04
0.064
109.94
0.073
109.14
0.073
109.8
0.073
111.12
0.082
110.35
0.082
111.07
0.082
112.16
0.091
111.39
0.091
112.13
0.091
145

5.3.3

Solution Controls
An axial load of 130 kips was applied on the top surface of the load stub for all the

analyzed test units. The magnitude of the distributed load was 0.9 ksi applied on a bearing
area of 8 in. x 18 in. to model the real test setup. A horizontal ramp load was applied on a
side surface of the load stub. The magnitude of the distributed load was decided according
to the load-carrying capacity of the test unit. The bottom surface of the footing was fixed to
the floor. Figure 5.1(a) shows the axial and horizontal loads with the bottom boundary
condition and the symmetric constraints.
5.3.4

Results
Figure 5.2 to Figure 5.5 show a comparison of the global force-displacement curves

between the experiment and the ABAQUS analyses for test units M05M, M10M, M15M and
M20M [54]. It can be seen that the simulated global behavior of the test units matched the
experimental response well. However, the models were not able to predict the web crushing
failure since no load-carrying capacity drop was observed from the simulation. The
comparison was made by plotting the numerical results up to the point same displacement
reached in the experiment. Further comparisons were made by investigating the flexure and
shear behavior separately. The same data reduction process was conducted on each model
with a complete set of numerical simulated data output corresponding to the experimental
instrumentation measurements. Figure 5.6 and Figure 5.7 show comparisons of the flexural
and shear force-displacement behavior for wall M05M, respectively; while Figure 5.8 and
Figure 5.9 show the corresponding comparisons for wall M20M. Although the global
behavior was well predicted, distinct discrepancies were observed on the predicted flexural
and shear components of behavior. It can be seen that shear deformations were over146

predicted. Actually, data reduction shows that the calculated shear deformation from the
bottom panel and the top panel was close for the four models, which indicates that the web
experienced uniform damage, as is observed on the contour plots for the compressive
damage index (DamageC). Figure 5.10 and Figure 5.11 show the contour plots of DamageC
for walls M05M and M20M.
On the other hand, the ABAQUS analyses under-estimated flexural deformations.
Figure 5.12 and Figure 5.13 show the comparisons of curvature profiles between the tests
and analyses. It can be seen that the spread of plasticity along the height of the wall was not
as large as in the experiment. Plasticity in the FE models was mainly concentrated in the
bottom 2 ft (610 mm) of the wall height. The comparison of curvature profiles for wall
M05M is better since the wall failed in elastic shear and the spread of plasticity barely
developed.
To evaluate the stress state in the web, especially at the web crushing region, Figure
5.14 and Figure 5.15 show the contour plots of compressive equivalent plastic strain
(PEEQ) for walls M05M and M20M. The maximum strain region is located next to the
compression boundary element and expands out with increasing displacement demand,
which is consistent with the web crushing regions observed in the tests of walls M05M and
M20M. However, it is difficult for the contour to reflect the crack pattern observed in the
experiment, such as the expansion of the diagonal tensile cracking in the web. This
shortcoming may be attributed to the CDP model, which is a continuum based model and
has no crack concept in its formulation.
It is of interest to evaluate the stress state of the elements in the web crushing region
in order to investigate the degradation of concrete strength under such complicated stress
state. Figure 5.16 and Figure 5.17 show traces for the minimum principal stress versus the
147

logarithmic strain of elements in the crushed region for walls M05M and M20M,
respectively. It is important to notice that the maximum compressive stress reached before
failure is close for both test walls, which indicates that HSC degrades significantly under
multi-axial stress state and only 10~20% of the uniaxial concrete compressive strength is left
while for the NSC wall, almost 50% of the strength is left at the onset of failure. It should be
noted that the curves of minimum principal stress versus logarithmic strain are not similar to
the uniaxial compressive stress-strain curves due to the highly localized behavior and
complicated loading and unloading that happens in the web-crushing region, even under a
monotonic loading process.

148

Ductility
0

1

2

3

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

3.5

Displacement (in.)
Figure 5.2 M05M global force-displacement behavior comparison between experiment and
ABAQUS.
Ductility
0

1

2

3

4

5

6

7

8

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0

1

2

3

4

5

0

6

Displacement (in.)
Figure 5.3 M10M global force-displacement behavior comparison between experiment and
ABAQUS.

149

Ductility
0

1

2

3

4

5

6

7

8

1000
800
150

600

100

400

50

200

Experiment
FEA (Abaqus)

0
0

1

2

3

4

5

Actuator Force (kips)

Actuator Force [kN]

200

0

6

Displacement (in.)

Figure 5.4 M15M global force-displacement behavior comparison between experiment and
ABAQUS.
Ductility
0

1

2

3

4

5

6

7

8

9

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0

1

2

3

4

5

6

7

0
8

Displacement (in.)

Figure 5.5 M20M global force-displacement behavior comparison between experiment and
ABAQUS.
150

Ductility
0

1

2

3

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

3.5

Displacement (in.)

Figure 5.6 M05M flexural force-displacement behavior comparison between experiment and
ABAQUS.
Ductility
0

1

2

3

1000
800
150

600

100

400

50

200

Experiment
FEA (Abaqus)

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Actuator Force (kips)

Actuator Force [kN]

200

0

3.5

Displacement (in.)

Figure 5.7 M05M shear force-displacement behavior comparison between experiment and
ABAQUS.
151

Ductility
0

1

2

3

4

5

6

7

8

9

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0

1

2

3

4

5

6

7

0

8

Displacement (in.)

Figure 5.8 M20M flexural force-displacement behavior comparison between experiment and
ABAQUS.
Ductility
0

1

2

3

4

5

6

7

8

9

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Abaqus)

0
0

1

2

3

4

5

6

7

0

8

Displacement (in.)

Figure 5.9 M20M shear force-displacement behavior comparison between experiment and
ABAQUS.
152

DAMAGEC
(Avg: 75%)
+8.900e-01
+8.158e-01
+7.417e-01
+6.675e-01
+5.933e-01
+5.192e-01
+4.450e-01
+3.708e-01
+2.967e-01
+2.225e-01
+1.483e-01
+7.417e-02
+0.000e+00

Figure 5.10 M05M contour plot: Compressive damage variable (DAMAGEC)
DAMAGEC
(Avg: 75%)
+9.340e-01
+8.562e-01
+7.783e-01
+7.005e-01
+6.227e-01
+5.448e-01
+4.670e-01
+3.892e-01
+3.113e-01
+2.335e-01
+1.557e-01
+7.783e-02
+0.000e+00

Figure 5.11 M20M contour plot: Compressive damage variable (DAMAGEC)
153

Height above footing, h [mm]

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2
y=4.606e-6 rad/mm

2000

Average curvature (x 10-3 rad/in.)
0.2

2000
1600
1200
800
400
0

0.3

0.4

0.5

0.6

0.7

0.8

90
1200
=5.827e-6 rad/mm
y
80
=1.15 (Experiment)
Average curvature (x10-3 rad/in.)
70
800
=1.67 (Experiment)
0.0 0.3 0.6 0.9 1.2 1.5 1.8 60
2.1 2.4 2.7 3.0 3.3
=1.15 (Abaqus)
90
400
50
=1.67 (Abaqus)
80
2000
y=4.606e-6 rad/mm
40

1600

70
0 10 20 30 40 50 60 70 80 90 100
60

1200

Average curvature [x10-6 rad/mm
50
10
=1
40

0

30
20

5

25

=2
30

Average curvature [x 10-6 rad/mm]

0

Height above footing, h (in.)

0.1

Height above footing, h [mm]

Height above footing, h [mm]

0.0

1600

0

=3

10
800

15

20

400

30

20

=4

Figure 5.12 M05M curvature profile comparison between experiment=5 ABAQUS.
and

0

=6
0 10 20 30 40 50 60 70 80 90 100 110 120

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 -6
Average curvature [x10 rad/mm]
=8 90
Experiment
FEA (Abaqus)
=1
=1 80
2000
1600

y=4.606e-6
(rad/mm)

=2
=3
=4

1200

=5
=6

800

=8

400

=2 70
=3 60
=4 50
=5 40
=6
=8

=1

0

=2

30
20
10

Height above footing, h (in.)

Height above footing, h [mm]

Average curvature (x10-3 rad/in.)

0

0 10 20 30 40 50 60 70 80 90 100 110 120

=3

Average curvature [x10-6 rad/mm]
=4
=1
=5
Figure 5.13 M20M curvature profile comparison between experiment and ABAQUS.
=2
=6
154
=3
=4
=5

=8

10
0

PEEQ
(Avg: 75%)
+1.343e-02
+1.231e-02
+1.119e-02
+1.007e-02
+8.953e-03
+7.834e-03
+6.714e-03
+5.595e-03
+4.476e-03
+3.357e-03
+2.238e-03
+1.119e-03
+0.000e+00

Figure 5.14 M05M contour plot: Compressive equivalent plastic strain (PEEQ).
PEEQ
(Avg: 75%)
+3.840e-02
+3.520e-02
+3.200e-02
+2.880e-02
+2.560e-02
+2.240e-02
+1.920e-02
+1.600e-02
+1.280e-02
+9.601e-03
+6.401e-03
+3.200e-03
+0.000e+00

Figure 5.15 M20M contour plot: Compressive equivalent plastic strain (PEEQ).
155

0.00
-3.0

-0.01

-0.02

-0.03
-20

-2.5

-16

-2.0
-12
-1.5
-8

-1.0

-4

-0.5
0.0
0.00

0
-0.01

-0.02

Stress Component, Min. Principal [MPa]

Stress Component, Min. Principal (ksi)

Logarithmic Strain Component, Min. Principal (m/m)

-0.03

Logarithmic Strain Component, Min. Principal (in./in.)
Figure 5.16 M05M: The min. principal stress-log. strain curve of a crushed element.

0.00
-3.0

-0.01

-0.02

-0.03

-0.04

-0.05
-20

-2.5

-16

-2.0
-12
-1.5
-8

-1.0

-4

-0.5
0.0
0.00

0
-0.01

-0.02

-0.03

-0.04

Stress Component, Min. Principal [MPa]

Stress Component, Min. Principal (ksi)

Logarithmic Strain Component, Min. Principal (m/m)

-0.05

Logarithmic Strain Component, Min. Principal (in./in.)
Figure 5.17 M20M: The min. principal stress-log. strain curve of a crushed element.
156

5.4 VecTor2 Analysis of Single Walls under Monotonic Loading
5.4.1

Geometrical Modeling
The single wall finite element models were created with Formworks, the pre-

processing software include in the VecTor2 Bundle v2.80 (Full Version) [39]. A single finite
element mesh is defined for all the analyses, as shown in Figure 5.18. The element size is
approximately 51 mm x 51 mm (2” x 2”). Five RC regions were defined: one region for the
footing, two regions for the boundary elements, one region for the web and one region for
the loading block. The regions are shown in different colors in Figure 5.18. The steel
reinforcement is smeared uniformly in the concrete regions and is not shown.

Figure 5.18 VecTor2 finite element mesh for single walls

157

5.4.2

Material properties
The smeared reinforcement properties were defined together with the definition of

concrete properties for each concrete type. There were four concrete types in total, one for
the footing, one for the boundary elements, one for the web and one for the loading block.
The material properties were defined based on the experimental data for each wall unit. For
the footing and loading block, a large value for the elastic modulus was defined to simulate
the footing and loading block as rigid regions.
Table 5-6 Summary of the material behavior models in VecTor2.
Convergence criteria
Compression base curve
Compression post-peak
Compression softening
Tension stiffening
Tension softening
Tension splitting
Confinement strength
Concrete dilatation
Cracking criterion
Crack shear check
Crack width check
Concrete bond
Concrete creep/relax
Concrete hysteresis
Steel hysteresis
Rebar dowel action
Rebar buckling
Previous load history
Slip distortion
Strain rate effects
Geometric nonlinearity
Crack allocation

Displacements - weighted
Popovics (NSC/HSC)
Popovics /Mander
Vecchio 1992-A
Modified Bentz
Linear
DeRoo 1995
Kupfer/Richart
Variable - Kupfer
Mohr-Coulomb (Stress)
Not considered
Crack limit (agg/5)
Eligehausen model
Not Considered
NL w/decay (Palermo)
Seckin model
Tassios (crack slip)
Not considered
Considered
Vecchio-Lai
Not considered
Not considered
Variable (Sato)

As previously mentioned, VecTor2 has a large library of phenomenological
constitutive models that permit realistic simulation of complex aspects in the response of
reinforced concrete structures. A summary of the models defined for the concrete regions,

158

the reinforcement and other behavioral models are shown in Table 5-6. For the compression
base curve, the M05M concrete was modeled with the Popovics normal strength concrete
(NSC) model, while the models for all other walls used the Popovics high-strength-concrete
(HSC) model.
5.4.3

Solution Controls
The applied boundary conditions are described here. The nodes at the bottom of the

footing block were constrained in their translational degrees of freedom. Two load cases
were defined to represent the axial and horizontal loading. Axial load was applied on top of
the load block, evenly distributed on the six nodes at the center of the block. The horizontal
load was applied in displacement control at the node on the left edge of the loading bock,
203 mm (8”) away from the top. Figure 5.19 shows the finite element mesh with the two
load cases shown as arrows and the constraints at the bottom shown as dark blocks.
5.4.4

Results
Figure 5.20 to Figure 5.23 show the comparisons of global force-displacement curves

between the experiment data and the results from the VecTor2 analyses for test units M05M,
M10M, M15M and M20M [54]. The M05M model is predicted by VecTor2 to fail earlier
than what took place in the real experiment. The early failure prediction from the analysis
may be due to the over-prediction of the initial stiffness. This is possibly because the tension
stiffening effect was over-predicted. With the increase of concrete compressive strength, the
effect was reduced and the global behavior was adequately simulated. The capacity of wall
M15M was over-predicted, probably due to the fact that the property of the #8 rebar in this
test unit was not defined appropriately due to the lack of the available test data.

159

Figure 5.19 Finite element model with load cases for single walls in VecTor2.

Another feature of the VecTor2 analyses on the single walls under monotonic
loading, which can be seen in Figure 5.20 through Figure 5.23, is that web crushing failure
was predicted appropriately by a drop of the load-carrying capacity. Figure 5.24 and Figure
5.25 show the numerically predicted deformed shape for walls M05M and M20M at web
crushing failure. For wall M05M, web crushing occurred next to the compression boundary
element along the height of the web, while for M20M, web crushing took place within the
plastic hinge region. It can be seen from the deformed shapes of the FE models that the
highly distorted (crushed) elements that the failure is mainly due to the shear deformation
induced on the elements. The high difference in stiffness between the boundary elements
and the web causes a tendency of vertical slip at the interface, where web crushing happens.
160

However, this effect is clearly more pronounced during elastic web crushing (M05M) than in
inelastic web crushing (M15M), in which the largest shear deformations (including those at
the interface between wall and boundary element) are concentrated within the plastic hinge
region Figure 5.26 and Figure 5.27 show the direction of the principal compressive strain at
failure. The parallel lines in Figure 5.26 indicate the elastic crack pattern in the web. In
Figure 5.27, crack realignment was observed in the web crushing region and the
compression zone of the boundary element is under high stress demand. This later behavior
is consistent with the observed fanning cracking mechanism observed in for the walls failing
by inelastic web crushing. The region of the web next to the tension boundary element is still
highly stressed possibly due to the localized deformation introduced by the tension boundary
element onto the web.
It is of interest to evaluate the stress state of the elements in the web crushing region
to investigate the degradation of concrete strength for elements under such a complicated
stress state. Figure 5.28 and Figure 5.29 show traces of the principal compressive stress
versus the principal compressive strain for walls M05M and M20M. It can be seen that
minor concrete strength degradation is predicted for the M05M wall element, which reached
a maximum compressions stress of approximately 4.6 ksi. By comparison, the HSC concrete
element in wall M20M experienced an appreciable amount of strength degradation and only
around 30% of its uniaxial strength is utilized. It should be noted that the residual stress of
the crushed elements at failure is only between 1 to 2 ksi. It also can be seen that for the
HSC model (wall M20M), the stress-strain curves of the crushed element displays unloading
and reloading behavior even though the global model loading was monotonic.

161

Ductility
0

1

2

3

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Vector2)

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

3.5

Displacement (in.)

Figure 5.20 M05M global force-displacement behavior comparison between experiment and
VecTor2.
Ductility
0

1

2

3

4

5

6

7

8

1000
200
150

600

100

400

Force (kips)

Force [kN]

800

50

200

Experiment
FEA (Vector2)

0
0

1

2

3

4

5

0

6

Displacement (in.)

Figure 5.21 M10M global force-displacement behavior comparison between experiment and
VecTor2.
162

Ductility
0

1

2

3

4

5

6

7

8

250

1000
200
150

600

100

400
200

50

Experiment
FEA (Vector2)

0
0

1

2

3

4

5

Force (kips)

Force [kN]

800

0

6

Displacement (in.)

Figure 5.22 M15M global force-displacement behavior comparison between experiment and
VecTor2.
Ductility
0

1

2

3

4

5

6

7

8

9
1000

200
150

600

100

400

50

200

Experiment
FEA (Vector2)

0
0

1

2

3

4

5

6

7

Force (kips)

Force [kN]

800

0
8

Displacement (in.)

Figure 5.23 M20M global force-displacement behavior comparison between experiment and
VecTor2.
163

Figure 5.24 M05M deformed model at web crushing failure.

Figure 5.25 M15M deformed model at web crushing failure.
164

Figure 5.26 M05M direction of principal compressive strain (ε2) at failure.

Figure 5.27 M15M direction of principal compressive strain (ε2) at failure.
165

-0.002

-0.004

-0.006

-0.008

-0.010

-0.012
32
28

-4

24
-3

20
16

-2

12
8

-1

4
0
0.000

-0.002

-0.004

-0.006

-0.008

-0.010

0
-0.012

[MPa]

0.000
-5

Concrete Net Stress Component,

Concrete Net Stress Component,

(ksi)

Concrete Total Strain Component, 2 (m/m)

Concrete Total Strain Component, 2 (in./in.)

Figure 5.28 M05M: The principal compressive stress-strain curve of a crushed element.

-0.005

-0.010

-0.015

-0.020
40

[MPa]

-6

0.000

35

-5

30

-4

25

-3

20

-2

15
10

-1

5

0
0.000

-0.005

-0.010

Concrete Total Strain Component,

-0.015

0
-0.020

Concrete Net Stress Component,

Concrete Net Stress Component,

(ksi)

Concrete Total Strain Component, 2 (m/m)

(in./in.)

Figure 5.29 M20M: The principal compressive stress-strain curve of a crushed element.
166

5.5 ABAQUS Analysis of Single Walls under Cyclic Loading
The analysis of the single walls under cyclic loading was attempted with ABAQUS.
However, the concrete damaged plasticity (CDP) model was unable to capture the unloading
and reloading behavior of the wall under cyclic loading, which is explained in [54] in detail.

5.6 VecTor2 Analysis of Single Walls under Cyclic Loading
5.6.1

General description
With respect to geometrical modeling, the same model as described in Section 5.4.1

was used for analyzing the walls under cyclic loading. Regarding material properties, the
same set of the concrete and steel material models as given in Table 5-6 were used for the
analysis here. The only differences were on the definition of the compression base curve.
Wall M05C was modeled with the Popovics normal strength concrete (NSC) model, while all
the other walls used the Popovics high-strength-concrete (HSC) model. The general
description of solution controls in Section 5.4.3 still applies to the cyclic analyses. The only
difference from the monotonic analyses is in the horizontal loading pattern. The definition
of the horizontal loading followed the loading protocol applied during test. First, four cycles
in force control were applied, followed by displacement-controlled cycles.
5.6.2

Results
Figure 5.30 to Figure 5.33 show comparisons of the global force-displacement curves

between experiment and VecTor2 analyses for test units M05C, M10C, M15C and M20C
[54]. The simulated M05C model failed on the first excursion to
specimen failed on the first excursion to

= -1, while the test

= 2 as is shown in Figure 5.30. It can be seen

that the load-carrying capacity of the wall at

= 1 is over predicted, which may be due to
167

two reasons. First, the initial stiffness of the model from VecTor2 is higher than the test
unit. Second, the definition of the steel properties for the #8 longitudinal rebar may have
not been appropriate due to the lack of experimental data. The simulated M10C model failed
on the second excursion to
the first excursion to

= 1.5 as is shown in Figure 5.31. The test specimen failed on

= 2. It is shown that the simulation captured the behavior of the

specimen even though the initial stiffness of the model was not well predicted and the
unloading stiffness of the model is a bit higher. For wall M15C, the test specimen failed on
the second excursion to
failure at the end of

= 4, but the simulated model has no indications of web crushing
= 4, as is shown in Figure 5.32. Wall M20C failed experimentally

upon reaching the first peak to
web crushing failure at the end of

= 6, but the simulated model showed no indication of
= 4, as is shown in Figure 5.33.

Overall, the global hysteretic behavior of the structural walls under cyclic loading was
well captured by the VecTor2 models. Web crushing failure of the NSC walls, like M05C
and M10C was predicted. Figure 5.34 shows the numerically predicted deformed shape the
M10C model at web crushing. Features of an elastic shear failure are observed. Figure 5.35
shows the numerical deformed shape for wall M20C at

= 4 x 2. No indication of web

crushing is observed throughout the applied cycles for the HSC wall modelss, which will be
discussed in Section 5.7.
Figure 5.36 and Figure 5.37 show curves for the principal compressive stress versus
the principal compressive strain of the crushed/critical elements in the models for walls
M05C and M20C. Since no web crushing was observed in the simulation of wall M20C until
the end of

= 4, Figure 5.37 shows the stress-strain curve of a critical element. That is, a

highly stressed or distorted element in the wall web adjacent to the compression boundary

168

element and within the plastic hinge region. The figures look different due to the distinct
difference in the strain range. The maximum compression stress reached by the walls under
cyclic loading was approximately 5 ksi, which is similar to the stress level reached by the wall
models under monotonic loading. Thus, the simulation shows no appreciable further
concrete strength degradation under cyclic loading based on the output of the stress-strain
curves.

169

Drift
-2% h

-1% h

800

0% h

1% h

2% h

3% h

Fy
Fy'

600

150
100

Force [kN]

400
200
0
-200

50

Ductility
-3

-2

-1

0

1

2

3

-400
-800
-75 -60 -45 -30 -15

0

0
-50
-100

-Fy'
-Fy

-600

200

Force (kips)

-3% h

Experiment
-150
FEA (Vector2)
-200
30 45 60 75

15

Displacement [mm]
Figure 5.30 M05C global force-displacement behavior comparison between experiment and
VecTor2.
Drift
-2% h

-1% h

800

0% h

1% h

2% h

3% h

Fy
Fy'

600

150
100

Force [kN]

400
200
0
-200

Ductility
-3

-2

-1
-1

0

1

2

3

-400
-800

50
0
-50
-100

-Fy'
-Fy

-600

200

Force (kips)

-3% h

Experiment
-150
FEA (Vector2)

-200

-75 -60 -45 -30 -15

0

15

30

45

60

75

Displacement [mm]

Figure 5.31 M10C global force-displacement behavior comparison between experiment and
VecTor2.
170

Drift
-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
800

Fy
Fy'

600

150

200

50

Ductility

0
-200 -5

-4

-2
-2

-3
-3

-1

1
-1

0

2

3

4

0

5

-50

-400
-600
-1000
-100

-75

-50

-25

-100

-Fy'

-Fy

-800

0

Force (kips)

100

400

Force [kN]

200

-150
Experiment
FEA (Vector2)
-200
50
75 100

25

Displacement [mm]
Figure 5.32 M15C global force-displacement behavior comparison between experiment and
VecTor2.
Drift

-4% h -3% h -2% h -1% h 0% h 1% h 2% h 3% h 4% h
800

Fy

600

Fy'

150

Force [kN]

200
-200

Ductility
-5

-4

-3

-2

-1

1

0

2
2

-400
-600

-Fy

-100

-75

-50

-25

0

4

5

0
-50
-100

-Fy'

-800

3

50

Force (kips)

100

400
0

200

25

Experiment
-150
FEA (Vector2)
-200
50
75 100

Displacement [mm]
Figure 5.33 M20C global force-displacement behavior comparison between experiment and
VecTor2.
171

Figure 5.34 M10C deformed model at web crushing failure.

Figure 5.35 M20C model with magnified deformation at
172

=4 x 2.

0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08

Concrete Net Stress Component,

30
-4
24
-3
18
-2

12

-1

6

0

0
0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08

[MPa]

-5

Concrete Net Stress Component,

(ksi)

Concrete Total Strain Component, 2 (m/m)

Concrete Total Strain Component, 2 (in./in.)

Figure 5.36 M05C: The principal compressive stress-strain curve of a crushed element.

0.000

0.001

0.002

0.003
-40
-35

-5

-30

-4

-25

-3

-20

-2

-15
-10

-1

-5

0
-0.001

[MPa]

-0.001
-6

(in./in.)

0
0.000

0.001

0.002

0.003

Concrete Net Stress Component,

Concrete Net Stress Component,

(ksi)

Concrete Total Strain Component,

Concrete Total Strain Component, 2 (in./in.)

Figure 5.37 M20C: The principal compressive stress-strain curve of a critical element.
173

5.7 Discussion
A data reduction process was conducted on the VecTor2 analytical results for the
M20C wall model. The displacement history of the nodes corresponding to the location of
displacement transducers on the test unit were extracted and post-processed in the same
procedure as for the experimental data. Separate flexure and shear behavior could then be
compared between the experiment and the VecTor2 results as shown in Figure 5.38 and
Figure 5.39. Good agreement was achieved between the experimental and numerically
simulated flexural and shear response of wall M20C. To investigate the difference in inelastic
shear behavior within the plastic hinge region and the elastic shear behavior in the region
above the plastic hinge, the force-shear displacement curves for the „numerically equivalent‟
bottom and top shear deformation panels for the FE simulation are shown in Figure 5.40. It
can be noted that the simulated response in the bottom panel exhibits inelastic shear
behavior while the numerical results for the top panel shows the elastic shear behavior. This
is consistent with the previously presented experimental data and attests to the adequate
simulation capabilities of VecTor2. Concrete degradation in the plastic hinge region due to
flexure effects is the main reason for inelastic shear behavior. The term „flexure-shear
interaction‟ is referred to describe this phenomenon herein.
To evaluate the ability of to capture the web crushing failure of wall M20C in
VecTor2, additional displacement-controlled cycles were applied. Web crushing was
predicted to happen on the second cycle of

= 8, see Figure 5.41. Another analysis was

done on the same M20C model with pure monotonic displacement-controlled loading. A
comparison of the hysteretic curves under cyclic and monotonic loading is shown in Figure
5.41. It can be seen that failure is predicted at the same ductility level for both models. Thus,

174

it can be seen that while stiffness degradation was adequately captured, degradation due to
localized damage due cyclic loading could not be captured by the VecTor2 analysis.

5.8 Conclusions
Modeling of single wall test units with 3D continuum plasticity-based models
(ABAQUS) and 2D plane stress phenomenologically-based models (VecTor2) was
conducted. The results were compared with the experimental data used to provide further
insight into web crushing behavior. Findings based on the analyses are as follows.
First, both ABAQUS and VecTor2 analyses were able to adequately capture the
global monotonic response of the single wall units. VecTor2 especially showed good
performance in modeling the cyclic and inelastic behavior of the HSC walls. The use of
ABAQUS and its CDP model on the analysis of single walls under cyclic loading was
unsuccessful. The CDP model needs to be modified to be able to capture degradation from
unloading and reloading stiffness under cyclic loading. Moreover, the issue of the viscoplasticity regularization needs to be clarified.
Second, the plasticity-based analyses cannot predict the web crushing failure.
Conversely, the phenomenological models built into VecTor2 were able to predict failure
under monotonic loading but not under cyclic loading. Further investigation is needed
regarding the modeling of web crushing capacity of structural walls under cyclic loading.
Lastly, the analytical results confirm that web crushing in the form of the inelastic
shear failure is caused by the flexure-shear interaction effects induced in the web of the
structural walls. The absolute values of minimum principal stress reached at the onset of
failure are predicted to be much lower than the nominal concrete compressive strengths,
usually around or below 5 ksi for the eight tested single wall units.
175

Drift
-3% h -2% h -1% h 0% h
800

2% h

3% h

Fy

600

1% h

Fy'

200
150
100

200

50

0
-200

Ductility
-4

-3

-2
-2

-1
-1

2

1

0

3

0

4

-50

-400
-600

-100

-Fy'

-800

-Fy
-80

-60

-40

-20

0

Force (kips)

Force [kN]

400

Experiment
-150
FEA (Vector2)
-200
40 60 80

20

Displacement [mm]
Figure 5.38 M20C flexural force-displacement curves from experiment and VecTor2.

Drift
-1% h

0% h

800

Fy

600

1% h

Fy'

150

Force [kN]

0
-200

50

Ductility
-1

0

0

1

-50

-400
-600

-100

-Fy'

-800

-Fy
-25 -20 -15 -10

-5

0

5

Force (kips)

100

400
200

200

Experiment
FEA (Vector2)
10

15

20

-150
-200

25

Displacement [mm]
Figure 5.39 M20C shear force-displacement curves from experiment and VecTor2.
176

Drift
-1% h

0% h

800

Fy

600

1% h

Fy'

150

Force [kN]

0
-200

50

Ductility
-1

0

0

1

-50

-400
-600

-100

-Fy'

-800

-Fy
-25 -20 -15 -10

-5

0

Force (kips)

100

400
200

200

Bottom Panel
Top Panel

5

10

15

20

-150
-200

25

Displacement [mm]
Figure 5.40 M20C VecTor2 analysis: comparison of the shear deformation between the
bottom and top panel.
Drift

-8% h -6% h -4% h -2% h 0% h 2% h 4% h 6% h 8% h
200

Fy

600

Fy'

150
100

Force [kN]

400
200
0
-200

50

Ductility
-10 -8

-6

-4

-2

2

0

4

6

8

10

-400
-600

-Fy

-200 -150 -100

-50

0

-50
-100

-Fy'

-800

0

Force (kips)

800

M20C Cyclic -150
M20C Mono. -200
50

100

150

200

Displacement [mm]
Figure 5.41 M20C global force-displacement behavior comparison under cyclic and
monotonic loading with VecTor2.
177

Chapter 6.
6.1

Simplified Analytical Method on HSC Structural Walls

Summary
This chapter presents the modification and use of a simplified inelastic analytical

model for determining the web crushing capacity of HSC structural walls. The strut-and-tie
web crushing model by Hines and Seible [16] is modified by the introduction of a new
calculation method for its concrete softening parameter. The analytical results show that the
modified Hines and Seible model can predict web crushing capacity of HSC structural walls
well. It has been shown that concrete degrades much faster in the HSC walls compared to
the NSC units . However, the modified approach to calculate the concrete softening
parameter was not able to capture further concrete strength degradation on wall M20C
compared to wall M15C, which caused the web crushing capacity of wall M20C to be overestimated. Nonetheless, the proposed modifications provide a more rational mechanism to
evaluate web crushing strength for HSC walls.

6.2 Concrete softening parameter
As is included in the compression field theory (CFT) and modified compression field
theory (MCFT), concrete compressive strength degrades with the coexistence of tensile
strains in the orthogonal direction. The equation for calculating the concrete softening
parameter

in the MCFT (1986) is:
(6-1)

In the model by Hines and Seible [16],

is calculated with the first strain invariant

quantity, which by means of Mohr‟s circle can be related to principal and transverse strains
by:
178

(6-2)
where

is the concrete principal tensile strain;

strain;

is the longitudinal strain;

is the concrete principal compressive

is the transverse strain. Since Hines‟ model [16] is

essentially a truss model, no shear transfer at the crack interface is considered. Thus,
the direction perpendicular to the strut and
is further simplified by leaving

is in

is in the direction of the strut. Equation (6-2)

out of the equation when

is assumed, as is done

in the development by Hines.
Hines proposes that the transverse strain,

can be obtained by using the UCSD

three component model [55] to calculate the stress in the transverse steel across the crack at
each force and displacement level in an averaged sense:
(6-3)
The transverse strain can then be calculated by assuming that the strain has not reach yield:
(6-4)
In the Hines and Seible model

is calculated as a constant quantity with an

empirical equation calibrated based on available test data. It is realized by Hines that the level
of transverse and longitudinal steel plays a significant role in the strength of the web
crushing region. Longitudinal and transverse reinforcement can help to reduce the crack
width and the principal tensile strain

. Thus, the equation for calculating

is inversely

proportional to the sum of the transverse and longitudinal reinforcement ratio in the web,
that is:
(

179

)

(6-5)

where

is the longitudinal reinforcement ratio in the wall and,

reinforcement ratio in the wall, and

is the transverse

is the yield strain of the longitudinal reinforcement.

Evidence that highlights the shortcomings to the approach by Hines in calculating
concrete softening parameter, and which motivate the modifications in this study follow.
First, it is reasonable to estimate the strains

and

at each force and displacement level,

analogous to the approach for calculating

based on the UCSD three-component model.

Howver, in the model by Hines and Seible

is computed based on the empirical Equation

(6-5) and kept constant, which does not comply with the requirement that the strain should
increase with the shift of the neutral axis under increased moment demand. Moreover, since
no test data on HSC structural walls was available for calibrating Equation (6-5), it may not
be appropriate to extend its applicability for the current study.
On the other hand, it seems reasonable to relate

to a moment curvature analysis,

which is at the core of the capacity model by Hines and Seible, and where

is a basic

output variable. In this way, the effect of HSC can be included since HSC walls have larger
rotating capacity by the reduction of the compression zone at higher moment demands.
Second, the strain of the transverse steel

in the model by Hines is calculated using

the UCSD three-component model on an averaged sense. The stress level of the steel is
directly related to the horizontal length of the crack, which is reflected in Equation (6-3) as
. For structural walls with highly reinforced boundary elements, the resultant tension and
compression forces are always located in the boundary elements, which will make

much

larger than the sectional depth of the web. However, boundary elements play an important
role in resisting tension and compression under moment and will mainly experience flexural
cracking. The cracking is mainly oriented in the horizontal direction as was observed in the
180

experiments in this study. It is also difficult for the diagonal shear cracking to penetrate and
crack the compression boundary element. It thus seems reasonable to limit the horizontal
length of the crack to the web depth only.
This evidence is assumed to be general and could be applied on any structural type
with heavily reinforced boundary elements or flanges. However, there is has no physical
meaning to say that the boundary elements do not resist shear demand and that the effective
cross sectional area can only be the web. The contributions of the boundary elements in
and

in the UCSD-three component are still admissible.
Third, the neglect of

one thing, neglecting
compressive strain

in Equation (6-2) could be problematic for two reasons. For

will result in an under-estimated value of

is negative. Based on the first order strain invariant or Mohr‟s circle,

should be equal to the sum of the absolute values of
assumption that

since the principal

,

and

. Additionally, the

is appropriate when tension stiffening effects are considered, as is

the case in finite element modeling. However, in a simplified truss model, the tension
stiffening effect is not considered. Also, since
at failure, a comparable value of

to

is the concrete principal compressive strain

could be possible.

Based on the preceding evidence, a modification is proposed to the model by Hines
and Seible in the approach to calculate the concrete softening parameter, as described below.
As a first modification, the distance between the resultant tension and compression
forces

in Equation (6-3) should be replaced with the cross sectional depth of the web

. Equation (6-3) thus becomes
(6-6)

181

This change is motivated by, and consistent with, the observed cracking pattern in the tested
walls in this study.
As a second change it is proposed that

be estimated based on the strain output of

the corresponding fiber from the moment-curvature analysis. Based on the geometry of the
strut in the inelastic shear cracking region, web crushing is expected to occur in the zone
next to the compression boundary element, which complies with the test observations of
NSC structural walls. For HSC walls, however, the web crushing region could be at a
different location. Since web concrete in the plastic hinge region is severely damaged, the
concept of a strut more of a theoretical assumption than the real test observation. Thus,
output the strain of the fiber at the interface of the web and compression boundary element
is believed to be reasonable.
It can be noticed that for compression struts that end at the compression boundary
element, the tip of the strut is above the base of the wall. Hence, the moment demand base
con which the strut demand is calculated should be scaled in proportion. The critical height
of the tip to the failing inelastic flexure/shear strut (

) is calculated as
(6-7)

The longitudinal strain of the fiber at the interface between the web and the compression
boundary element can be obtained from the moment-curvature analysis at the a moment
value scaled by

⁄ , where

is the height of the wall.

The third step in the modification is to determine the principal tensile strain
consistently with the relationships from Mohr‟s circle,
(

√(

)

182

)

(6-8)

where
strain,

is the concrete shear strain of the strut in the plastic hinge region. The shear
may be considered to be averaged over the wall and calculated as
(6-9)

where

is the shear deformation at the top of the wall, and
It should be noted shear strain

is the total height of the wall.

is the net strain of the concrete in the strut, which

does not include the shear deformation of the member due to the crack shear slip. Thus, the
estimate based on Equation (6-8) may be considered appropriate in an averaged sense.

6.3 Results
The web crushing capacity of the single walls tested under cyclic loading was
calculated based on the model by Hines and Seible [16] with the modifications on the
concrete softening parameter as described in Section 6.2. Figure 6.1 to Figure 6.4 show the
hysteretic behavior of the single walls under cyclic loading with the failure predictions. Table
6-1 shows a comparison of the inelastic web-crushing capacities of single walls under cyclic
loading with the Modified Hines-Seible model. The prediction quality of the Modified
Hines-Seible model for HSC structural walls improved dramatically, compared to the
original model. The experimental investigation showed that M20C was not able to attain
additional deformation capacity compared to M15C. Thus, it a good prediction of the
experimental response with the presented model is not possible since the model inherently
considers that that web crushing capacity is linearly related to concrete compressive strength.
The shear capacity curves based on Equation (2-10) of ACI 318-08 [25], Equation
(2-15a) by Oesterle et al. [4], Equation (2-16) by Paulay and Priestley [7] and the UCSD
three-component model [55] are also shown in Figure 6.1 to Figure 6.4. It should be noted
that for all the models, except for that from ACI 318-08, a maximum shear stress of
183

was used to avoid unreasonably high values at: (i) small interstory drift ratios for Oesterle‟s
model, (ii) small displacement ductility for Paulay and Priestley‟s model, and (iii) small
cracking angle and concrete softening values in Hines‟ model. The ACI 318-08 and the
UCSD three-component model are diagonal tension capacity models, which are shown along
the web crushing capacity models for comparison. Further, it should be noted that shear
strength plotted for the ACI 318-08 model is merely proportional to √
of

without the limit

psi.
It is observed that the limit of

does not control web crushing capacity of the

walls for concrete compressive strength above 6 ksi. However, it controls the prediction for
elastic web crushing failures, as was the case for the M05C wall. The noted limit thus
represents an upper bound and thus considered quite reasonable.

Displacement [in.]
0

1

2

3

4

5

1200

300
250
200

UCSD

800
600

O&al.

400

P&P

150

H&S

100

M-H&S

ACI

200

50

0

0
0

25

50

75

100

125

Displacement [mm]
Figure 6.1 M05C hysteretic behavior with failure predictions.

184

Force (kips)

Force [kN]

1000

Displacement [in.]
0

1

2

3

4

5

1200

800

UCSD

200

H&S

P&P

150

O&al.

600

M-H&S
100

ACI

400

Force (kips)

250

1000
Force [kN]

300

50

200
0

0
0

25

50

75

100

125

Displacement [mm]
Figure 6.2 M10C hysteretic behavior with failure predictions.

Displacement [in.]
0

1

3

4

5

300

H&S
UCSD

1000

P&P

800

250

O&al.

200

M-H&S

150

600
ACI

400

100
50

200
0

0
0

25

50

75

100

125

Displacement [mm]
Figure 6.3 M15C hysteretic behavior with failure predictions.
185

Force (kips)

1200

Force [kN]

2

Displacement [in.]
0

1

2

3

4

5
300

1000
Force [kN]

H&S

UCSD

250

P&P

M-H&S

800

O&al.

600

200
150

ACI

Force (kips)

1200

100

400

50

200
0

0

0

25

50

75

100

125

Displacement [mm]
Figure 6.4 M20C hysteretic behavior with failure predictions.
Table 6-1 Comparison of inelastic web-crushing capacities of single walls under cyclic
loading with Modified Hines-Seible model.

(mm)

(kN)

Modified Hines-Seible
Model
Diff.
(%)
(mm)
(kN)

M05C

45.0

803

48.5

821

8

M10C

42.7

751

54.9

740

29

M15C

78.7

819

77.2

814

2

M20C

76.5

815

93.9

922

23

Test
Unit

Experiment

Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN.

6.4 Discussion
An expanded discussion based on the results presented in Figure 6.1 to Figure 6.4
and the analytical data is provided below.
186

Figure 6.5 shows the concrete compression softening parameter at web crushing
failure calculated for the single walls under cyclic loading based on the original Hines-Seible
model and the Modified Hines-Seible model. The symbols in the figure represent the
calculated values of the parameter at failure, while the line trance is the analytical relation for
the modified approach. It can be seen in Figure 6.5(a) that the original Hines-Seible model
estimates for the softening parameter do not vary much for the different walls. The
softening parameter was thus insensitive to compressive strength and the degradation of
HSC under cyclic loading is not considered appropriately. In contrast, Figure 6.5(b) shows
that smaller values for the softening parameter were obtained for HSC walls compared to
the values for NSC walls. However, the values of the parameter are much larger than what
was obtained from the NLFEM modeling in Chapter 5, where the concrete under cyclic
loading reached maximum compression stress of only approximately 5 ksi based on the
VecTor2 analyses. It is believed that the difference resides in the geometry of the studied
objects. In the Modified Hines-Seible model, the free body diagram of the strut in the
inelastic shear cracking region was studied as the free body diagram, while in the NLFEM
study the stress values are obtained form an element, which represents very local behavior.
From Equation (6-1), it can be seen that the concrete compression softening
parameter is only a function of the principal tensile strain

, the effect of HSC is not

considered explicitly. Since the equation is calibrated based on the membrane tests on NSC,
the equation may not be appropriate for evaluating the strain of HSC. A calibration of the
equation is needed for better compliance with HSC members.
The shear strength limit of

√

in ACI 318-08 uniformly under-estimated the

web crushing capacity of structural walls in this study. Moreover, the equation cannot give

187

any valid information on the displacement ductility of the structural walls, which is an
essential index for the seismic design of the structural walls or bridge piers.
While Oesterle‟s model under-estimated web crushing capacity for wall M05C it gave
a good prediction on the other walls, which is not surprising after considering the model is
based on test results of similar structural walls failing in web crushing up to displacement
ductility 4. The prediction is not conservative for wall M20C since the model is linearly
related to the concrete compressive strength and no enhancement in ductility was observed
with concrete strengths over 15 ksi. On the other hand, the model by Paulay and Priestley
under-estimates the web crushing capacity for all walls, which shows that the model is
conservative enough for design. It should be noted that the predictions noted (and plotted)
for Oesterle‟s and Paulay-Priestley‟s models were modified from their original proposed
values as previously discussed. Thus, their performance, as discussed above, essentially
applies to a modified version of the models.
It must be realized that the Modified Hines-Seible model as well as the other shear
capacity modes were developed to predict web crushing capacity of structural walls under
cyclic loading only. There was no intent in predicting web crushing capacity for walls tested
under monotonic loading for the following reasons. The monotonic wall tests were
conducted for the purpose of serving as a reference to investigate the effect of cyclic loading
on concrete degradation. Moreover, the HSC walls under monotonic loading sustained large
inelastic deformations before web crushing, which was mainly due to the superior shear
transfer mechanism along cracks with the loading in one direction. The behavior of the
monotonically tested walls is thus valuable only for research purposes and is not deemed to
be realistic for design practice.

188

1.2

(a)

M05C
M10C
M15C
M20C

1.0
0.8
0.6
0.4
0.2
0.0
0.000

1.2

0.005

0.010

0.015

(b)

0.020

M05C
M10C
M15C
M20C

1.0
0.8
0.6
0.4
0.2
0.0
0.000

0.005

0.010

0.015

0.020

Figure 6.5 Concrete compression softening parameter at web crushing failure for single walls
under cyclic loading: (a) Original Hines-Seible model; (b) Modified Hines-Seible model.

189

6.5 Conclusions
In this study, a simplified analytical method was conducted to predict the web
crushing capacity of HSC walls. Modifications to the Hines-Seible model were proposed and
shown to dramatically improve the quality of web crushing predictions. The original method
for calculating concrete softening parameter was discussed and modifications were proposed
based on well-founded arguments. The main findings are described in the following.
First, the Modified Hines-Seible model is still based on the assumption that web
crushing capacity is linearly proportional to concrete compressive strength. This assumption
was verified in this study up to concrete compressive strengths of 15 ksi. However, test
result for wall M20C indicated that there is no appreciable improvement in ductility with
concrete strength of 20 ksi. The reason is attributed to the fact that the increased brittleness
of concrete with increased compressive strength makes it more susceptible to degradation
under cyclic loading. This effect is not captured in the original or modified model.
Therefore, the web crushing capacity of M20C wall was over-predicted.
Second, the Modified Hines-Seible mode is essentially a truss model. Different from
a NLFEM analysis, the complicated shear stress-distortion relationship at the crack interface
and the corresponding structural behavior are neglected. Concrete softening is thus still in an
averaged sense and it is different from what was obtained from the NLFEM analyses, which
represents a very local behavior.

190

Chapter 7.
7.1

Modeling and Analysis of HSC Hollow Bridge Piers

Summary
This chapter presents nonlinear finite element modeling and simplified analysis of

the Diagonal Pier Test (DPT) unit subjected to monotonic to study the three-dimensional
behavior of wall assemblies and to propose an approach to assess crushing behavior .
Nonlinear finite element analyses were conducted using 3D continuum plasticity-based
models in the program ABAQUS. The web crushing capacity of the HSC hollow bridge
piers was assessed with a simplified analytical method. In the simplified method, an
equivalent single wall loaded under in-plane demands in the diagonal direction of the cross
section is idealized. The web crushing capacity of the equivalent wall was analyzed with the
Modified Hines-Seible model. Analytical results show that diagonal web crushing capacity of
the DPT can be well represented by the equivalent single wall concept. However, the web
crushing capacity of the BPT unit is over estimated when using this modeling approach due
to the failure to consider the damage on concrete from multi-directional loading effects.

7.2 Diagonal Web Crushing Capacity
The web crushing capacity of wall assemblies, or piers as the ones tested in this
study, along their principal directions can be studied in the same way as single walls since the
wall assembly can be simplified as dual single walls with the flexural contribution of the two
webs in the orthogonal direction to loading being neglected. However, the web crushing
capacity of wall assemblies or hollow piers along non-principal directions requires different
considerations. Further, depending on the geometry of the walls in the assembly, the web
crushing capacity of the assembly (or pier) in non-principal directions may control the
capacity of the assembly for the following reasons. The aspect ratio of the individual walls in
191

the assembly about the diagonal direction is smaller than that in the principal directions. For
the hollow bridge piers test units in this study, the aspect ratio in diagonal direction is 1.94
compared to the aspect ratio of 2.5 in the principal directions. Figure 7.1 shows the analytical
force-displacement behavior curves of the Biaxial Pier Test (BPT) unit in the principal and
diagonal directions. It can be seen that the capacity of the pier about a diagonal flexural axis
is much higher load capacity but a reduced deformation capacity. It is thus anticipated that
web crushing will happen earlier in the diagonal direction than in the principal direction. In
this study, the diagonal web crushing capacity the hollow pier tested in the diagonal direction
(DPT) was investigated to characterize the three-dimensional web crushing capacity of the of
wall assemblies.

Displacement (in.)
1400

0

2

4

6

8

10

12

14
300

1200

250
200

800

150

600

100

400
200

Principal direction
Diagonal direction

0
0

50

100

150

200

250

300

350

Force (kips)

Force [kN]

1000

50

0
400

Displacement [mm]
Figure 7.1 The analytical force-displacement responses in principal and diagonal directions.

192

7.3 NLFEM of the Diagonal Pier Test Unit
The nonlinear finite element modeling (NLFEM) of the diagonal pier test (DPT)
unit was conducted with the general purpose finite element program ABAQUS. Since the
aim of the analysis was to study the diagonal web crushing behavior of the model and the
modeling capabilities in ABAQUS cannot adequately capture the desired cyclic degradation,
a monotonic ramp load was applied in the diagonal direction of the DPT unit. The overview
of the model is shown in Figure 7.2(a). Figure 7.2(b) shows the discrete reinforcement
defined for the model. The footing block was modeled as elastic concrete material and no
reinforcement was explicitly modeled in this region. The blocks of the column and the
footing are meshed separately with a tie constraint at the interface to connect them together.
The mesh for the concrete parts is shown in Figure 7.2(c).An embedment constraint was
applied between the reinforcement and the concrete blocks. The concrete damaged plasticity
(CDP) material model in ABAQUS was used for the concrete in the column body. Threedimensional eight-noded continuum-type solid elements with reduced integration (C3D8R)
were used for all concrete parts and three-dimensional two-noded truss elements with full
integration (T3D2) were used to model the steel reinforcement.
The material properties were defined based on test data from the DPT unit. Details
on the steps required to define the CDP model are given in Section 5.3.2. A summary of the
material parameters for the material models for DPT simulation is shown in Table 7-1.
To apply a monotonic ramp load on the DPT unit in the diagonal direction, two
force components (in x and y directions) were applied on the model. The horizontal force
components and the axial load are shown in Figure 7.2(a).

193

Figure 7.2 Nonlinear finite element model of diagonal pier test (DPT) unit: (a) Model
overview; (b) Reinforcement; (c) Mesh of concrete parts

Figure 7.3 shows a comparison of the global force-displacement behavior between
the experiment and the FE simulation. It can be seen that the global force-displacement
behavior of the DPT unit was predicted well. However, no failure (i.e., drop on the loadcarrying capacity) was captured, which indicates that the web crushing capacity of the test
unit is unable to be predicted through presented nonlinear finite element modeling
approach. Moreover, the effect of the visco-plasticity parameter for controlling the
convergence and the accuracy of the results need to be further investigated.

194

Table 7-1 The concrete and steel material parameters for DPT Unit.
The parameters of CDP model
27°
DPT

0.1
Concrete elasticity
1.16
E(ksi)
0.667
4578

0.0075
0.2

Concrete compression hardening
Concrete compression damage
Stress (ksi)
Crushing strain
DamageC
Crushing strain
5.81
0
0
0
7.24
0.00017
0.07
0.00017
8.00
0.00058
0.21
0.00058
5.26
0.00185
0.52
0.00185
1.71
0.00363
0.79
0.00363
0.57
0.00487
0.89
0.00487
0.28
0.00569
0.92
0.00569
0.15
0.00647
0.95
0.00647
Concrete tension stiffening
Concrete tension damage
Stress (ksi)
Cracking strain
DamageT
Cracking strain
0.90
0
0
0
0.66
0.00029
0.27
0.00029
0.45
0.00065
0.50
0.00065
0.26
0.00116
0.71
0.00116
0.16
0.00157
0.81
0.00157
0.08
0.00219
0.90
0.00219
0.03
0.00300
0.96
0.00300
Reinforcing Steel Properties
#6 Steel
#3 Steel
E(ksi)
29000
E(ksi)
29000
0.3
0.3
Stress (ksi)
Plastic strain
Stress (ksi)
Plastic strain
64.86
0
63.64
0
65.10
0.00374
63.77
0.00204
70.36
0.00752
71.07
0.00750
81.54
0.01699
81.82
0.01698
90.25
0.02645
90.36
0.02644
96.84
0.03588
96.95
0.03588
101.67
0.04528
101.91
0.04528
105.11
0.05464
105.50
0.05463
107.49
0.06395
108.04
0.06393
109.14
0.07320
109.80
0.07317
110.35
0.08237
111.07
0.08235
111.39
0.09147
112.13
0.09144

195

Figure 7.4 and Figure 7.5 show contour plots of PEEQ and PEEQT at a
displacement level of ductility of 4. It is shown that the diagonal truss mechanism in the
webs next to the compression boundary element is not well established to transfer the forces
from the boundary elements in tension to the compression boundary element. It seems that
the cross section of the model is mainly under flexural demands, which will result in tension
and compression forces only. The effect of shear demand seems to be isolated from the
flexural demand effects. It follows that the inability of the modeling approach to capture the
flexure-shear effects does not allow the model to display the diagonal truss mechanism in the
webs. Further investigation is needed in this regard.
Figure 7.6 and Figure 7.7 show contour plots for the shear stresses S13 and S23 on
the concrete parts of the model. Observation of the contour plots indicates that the webs
next to the compression boundary element carry more shear stresses compared to the webs
between the extreme tension boundary elements and the middle boundary elements. This
verifies that the shear carrying-capacity is reduced due to the occurrence of flexure cracking
in the tension walls, even though dowel action and shear transfer across cracks are still able
to resist shear. At the same time, the compression zone of the cross section, as well as the
region under less tension forces, can resist more shear stresses. This phenomenon exists on
coupled shear walls as well as other types of wall assemblies.

196

Drift at the inflection point

2000
1600
1200
800
400
0
-400
-800
-1200
-1600
-2000

500

Fy

400

Fy'

300
200
Ductility

-5

-4

-3

-2

-1

1

0

-Fy'
-Fy
-5

-4

-3

-2

-1

0

1

2

3

4

5

100
0
-100
-200

-300
Experiment
-400
FEA (Abaqus)
-500
2
3
4
5

Actuator diagonal force (kips)

Actuator diagonal force [kN]

-4% h-3% h-2% h-1% h 0% h 1% h 2% h 3% h 4% h

Diagonal displacement at the inflection point (in.)
Figure 7.3 DPT: global force-displacement behavior comparison between experiment and
ABAQUS.
PEEQ
(Avg: 75%)
+3.680e-02
+3.373e-02
+3.067e-02
+2.760e-02
+2.453e-02
+2.147e-02
+1.840e-02
+1.533e-02
+1.227e-02
+9.200e-03
+6.133e-03
+3.067e-03
+0.000e+00

Figure 7.4 DPT Unit contour plot: Compressive equivalent plastic strain (PEEQ).
197

PEEQT
(Avg: 75%)
+6.943e-02
+6.364e-02
+5.786e-02
+5.207e-02
+4.629e-02
+4.050e-02
+3.472e-02
+2.893e-02
+2.314e-02
+1.736e-02
+1.157e-02
+5.786e-03
+0.000e+00

Figure 7.5 DPT Unit contour plot: Tensile equivalent plastic strain (PEEQT).
S, S13
(Avg: 75%)
+5.273e+00
+4.705e+00
+4.137e+00
+3.569e+00
+3.001e+00
+2.433e+00
+1.865e+00
+1.297e+00
+7.289e-01
+1.608e-01
+4.072e-01
+9.753e-01
+0.000e+00

Figure 7.6 DPT Unit contour plot: shear stress component in horizontal plane (S, S13).
198

S, S23
(Avg: 75%)
+4.614e+00
+4.122e+00
+3.631e+00
+3.140e+00
+2.649e+00
+2.158e+00
+1.666e+00
+1.175e+00
+6.841e-01
+1.930e-01
+2.982e-01
+7.894e-01
+1.281e+00

Figure 7.7 DPT Unit contour plot: shear stress component in horizontal plane (S, S23).

7.4 Equivalent Single Wall Unit Analysis
Without considering the multi-directional loading effects, the response of both the
Diagonal Pier Test (DPT) and Biaxial Pier Test (BPT) units could be simplified as an
equivalent system under in-plane demands. This, rather large, assumption makes it possible
to analyze the diagonal web crushing behavior with an equivalent single wall. Figure 7.8
illustrates the elevation of such an equivalent single wall as a projection of the cross section
of the hollow pier test units. The performance of the Modified Hines-Seible model in
assessing the web crushing capacity of the pier units by means of a simplified equivalent wall
was assessed and described next.

199

Figure 7.8 Equivalent single wall concept for the capacity analysis of hollow piers.

The boundary elements, which are mainly designed to resist moment demands with
concentrated tension or compression forces, are not affected by being located spatially away

200

from the diagonal axis of bending. Thus, they can be assumed to be on the diagonal axis of
in-plane with the same dimensions. Unlike the web of a single cantilevered wall under inplane demands, the concept of an equivalent single wall for the pier wall assembly is such
that the web of the equivalent single wall is separated into two segments due to the existence
of the boundary elements in the middle. The effect of the two middle boundary elements is
described as follows.
First, separation of the middle boundary elements from the one in the far tension side
changes the location of the resultant tension force, which will be in the wall and move
towards the middle boundary elements with an increase in moment demand. Second, the
boundary elements divide the wall into two halves and disturb the cracking pattern. The
cracking pattern on each wall will be different in terms of cracking angles and shear demand.
Finally, the depth of the web for each segment is determined by the diagonal distance
between the adjacent boundary elements. The width of the equivalent wall was assumed to
be the width of each wall in the piers. This geometric modification can be obtained through
simple calculations based on the premise that the shear strength of the concrete must be the
same.
The web crushing capacity of the equivalent single wall was evaluated with the
different capacity models presented in Chapter 6. The Modified Hines-Seible model with the
concrete softening parameter as described in Section 6.2 was also used. There is a slight
difference on the depth of the web

in Equation (6.6) for estimating the stress of the

transverse steel. Due to the presence of boundary in the middle of the cross section of the
equivalent wall, the wall is separated and the depth is assumed to be reduced. This follows
from the fact that it is appropriate to assume there is only horizontal cracking in the middle
boundary elements due to the moment demand, the dimension of which should not be
201

included in

. The net depth of the web for the equivalent single wall for the pier test units

was approximately 734 mm (28.9”). With respect to the determination of

, the longitudinal

strain of the fiber next to the compression boundary element for the Modified Hines-Seible
model, it was noticed that the neutral axis is located in the web of the equivalent wall up to
web crushing failure. Therefore, the values of

were negative.

Figure 7.9 and Figure 7.10 show the hysteretic behavior for the DPT and BPT units
together with web-crushing capacity curves from different models applied to the just
described equivalent single wall. It can be seen that the web crushing capacity of the DPT
unit is well predicted by the Modified Hines- Seible model. However, the model is unable to
assess (over-predicts) the web crushing capacity of the BPT unit, whose capacity was
severely curtailed due to the multi-directional loading. The values of concrete softening
parameter

using the modifications presented in Section 6.2 were 0.77 for the DPT unit and

0.70 for the BPT unit. The reason that the Modified Hine-Seible model can not predict the
web crushing capacity of the BPT unit is that concrete damage due to the multi-directional
loading failed to be considered. It can be reasonably expected that the BPT pier test unit
would have been able to sustain a larger level of displacement ductility if the same loading
protocol of the DPT unit had been applied.

202

Displacement [in.]
1

2

1200
P&P

Force [kN]

1000

3

4

5

UCSD

M-H&S

250
O&al.

200

ACI

800

300

150

600

100

400

Force (kips)

0

50

200
0

0
0

25

50

75

100

125

Displacement [mm]
Figure 7.9 DPT Unit hysteretic behavior with failure predictions.

Displacement [in.]
0

1

2

3

1800

UCSD

1600

5

6

M-H&S

P&P

1400

Force [kN]

4

O&al.

400
350
300

ACI

1200

450

250

1000

200

800

150

600
400

100

200

50

0

0
0

25

50

75

100

125

150

Displacement [mm]
Figure 7.10 BPT Unit hysteretic behavior with failure predictions.
203

Force (kips)

2000

7.5 Conclusions
Nonlinear finite element simulations were conducted on the DPT unit with a
monotonic loading along the diagonal direction of the cross section. The modeling approach
was able to predict the global force-displacement behavior of the unit quite well. However,
the flexure-shear interaction behavior and the truss mechanism in the webs could not be
captured with the modeling approach followed in this work. Observations on the
distribution of the shear stresses indicate that the webs in the compression side carried more
shear stresses than the webs between the tension boundary elements. …
The diagonal web crushing capacity of the DPT and BPT units was also analyzed in
this chapter through an equivalent single wall concept. It was found that the Modified
Hines-Seible model presented in Section 6.2 predicted the capacity of the DPT pier well, but
over-predicted the capacity of the BPT unit. The reason behind over-predicting the web
crushing capacity of the BPT pier unit is attributed to the deficiency of considering concrete
damage due to the multi-directional loading effects. A parameter reflecting the loading
protocol effect is recommended to be included in calculating the reduced concrete
compressive strength.

204

Chapter 8.

8.1

Summary and Conclusions

Summary
High-strength-concrete (HSC) offers the potential of optimizing structural design

and reducing material costs. However, the using of HSC in seismic regions is not well
studied and corresponding design guidelines are deficient. Web crushing failure has long
been regarded as a brittle failure mode. The current design codes in the US avoid it by
limiting the maximum allowable stress provided by the reinforcement. Under such
circumstances, this dissertation has evaluated the web crushing performance limits of HSC
structural walls and wall assemblies. The hypothesis that HSC can lead to ductile shear
failures in reinforced concrete structural walls by delaying (i.e., increasing) web crushing
capacity was verified through a comprehensive experimental program and supported by
numerical and analytical investigations.
A single wall test program was conducted to verify the effect of concrete
compressive strength and cyclic loading on web crushing capacity. Eight 1/5-scale single
walls were tested with design concrete compressive strengths of 34, 69, 103, and 138 MPa (5,
10, 15 and 20 ksi) under cyclic and monotonic loading protocols. The experimental results
revealed that high-strength-concrete can effectively delay web-crushing failures and increase
the displacement ductility levels of structural walls. Contrary to the traditional opinion of
shear failures having rapid strength and stiffness degradation and less energy-dissipating
mechanism, “ductile shear failures” were identified on HSC structural walls. This behavior is
deemed to happen on structural walls with boundary elements or flanges where flexure-shear
interaction can take place in well-constrained thin-webbed elements. The flexure effect was
205

introduced in the wall webs by extension of the horizontal cracking in the tension-resisting
boundary element, which results in a fanning crack pattern in the wall web. On one hand,
the flexural-shear effect improves the deformation capability of the web before web
crushing. Yet, this mechanism causes strength and stiffness degradation of the concrete in
the web due to excessive cracking, which is the reason of web crushing, particularly from
cyclic loading. For high-strength-concrete with compressive strengths over 15 ksi, the
further improvement in ductility could be compromised by the rapid strength degradation
under cyclic loading.
A wall-assembly test program was conducted within the context of hollow bridge
piers to investigate the effect of three-dimensional demands on the web crushing capacity of
wall assemblies loaded in non-principal directions and under severe multi-directional loading
paths. Two 1/4-scale hollow rectangular bridge pier test units featuring highly-reinforced
boundary elements at the four corners and thin connecting webs, were subjected to diagonal
and multi-directional cyclic loading with design concrete strengths of 34 and 138 MPa (5 and
20 ksi), respectively. Experimental results showed that both test units exhibited stable ductile
behavior until web crushing at moderate ductility levels. The achieved comparable ductility
levels of the two pier test units indicated the advantageous effect of HSC in increasing web
crushing capacity is compromised by concrete damage and strength degradation under multidirectional loading. Evaluation of the diagonal web crushing behavior is complicated due to
the occurrence of middle boundary elements. Different cracking patterns were observed on
the diagonally load pier test unit (DPT) between the „compression‟ and „tension‟ walls, which
also indicates the effect of non-principal, or three-dimensional, demands on shear stress
distribution. However, for the biaxially loaded pier (BPT) the difference in cracking pattern
was minor due to the severity and dominance from the multi-directional loading effects.
206

Nonlinear finite element modeling (NLFEM) of web crushing capacity of highstrength-concrete structural walls was conducted with 3D continuum plasticity-based finite
element models (ABAQUS) and 2D plane stress phenomenologically-based finite element
models (VecTor2). Capabilities and limitations of both modeling approaches in evaluating
the web crushing failure mechanism and capacity of reinforced concrete walls were
investigated. Continuum-based FE analyses with concrete damaged plasticity modeling were
able to adequately simulate global behavior of the single walls under monotonic loading.
However, issues related to the numerical technique of viscoplastic regularization and the
simulation of stiff structural walls under cyclic loading were not clarified. Modifications to
the concrete damaged plasticity model in ABAQUS are deemed necessary. The
phenomenlolgically-based FE analyses were able to successfully model the behavior of
structural walls under both monotonic and cyclic loading. The models were even able to
capture the inelastic shear behavior in the plastic hinge region. However, concrete strength
degradation under cyclic loading and the web crushing capacity is not considered in the
formulation of the analytical model and the web crushing capacity of walls under cyclic
loading could not be predicted. Both simulation methods revealed that the concrete stress at
web crushing was only a small portion of the cylinder compressive strength, which is
attributed to the complex stress state of the concrete in the web crushing region. The
simulation results confirm that web crushing in the form of inelastic shear failure is caused
by the flexure-shear interaction effects induced in the web of the structural walls.
To evaluate web crushing capacity with simplified analytical methods, the inelastic
web crushing capacity model by Hines and Seible was used with modifications to the
calculation of the concrete softening parameter. The modifications were verified to be
appropriate to predict the capacity of HSC walls. It was noted through the analysis of the
207

single walls in this study that the HSC walls displayed more rapid concrete softening
compared with the NSC walls. To apply the simplified method to the pier test units, an
equivalent single wall concept about the non-principal loading direction of the cross section
was proposed. The modified Hines-Seible model was shown to accurately predict the web
crushing capacity of the DPT pier. However, the web crushing capacity of the BPT unit was
over estimated by this simplified analysis approach due to the failure to consider concrete
degradation from the effects of multi-directional loading.

8.2 Original Contribution
Contrary to current practice, which strictly avoids web crushing failures by
prescriptive requirements, this study has fundamentally addressed the issue. Instead of
simply being considered a brittle failure, web crushing in a structural wall can be designed to
occur after significant and reliable ductile performance is achieved by the use of highstrength-concrete. This counter-intuitive and non-traditional concept was verified through
an extensive and original experimental investigation on single in-plane loaded walls and wall
multi-directionally loaded hollow piers. Both experimental phases demonstrated that ductile
shear failure behavior with the application of high-strength-concrete can be designed for and
reliably predicted. The findings from this study can be extended to conclude that ductile
shear failure can happen on thin-webbed elements constrained by well-confined boundary
elements where flexure-shear interaction is introduced. The findings from this work provide
further evidence that the common design-code notion that web crushing is proportional to
the tensile strength of concrete and not dependent of the elements inelastic deformation is
not only incorrect but leads to over conservative estimate of behavior.

208

8.3 Future Research Needs
High-strength-concrete shows great potential for slender structural design by
increasing the web crushing capacity of structural walls. However, the brittle nature of HSC
can result in excessive cracking, which may not satisfy the functional requirements even if
safety is not the issue. The structural walls tested in this study shared the same geometry and
reinforcement ratios. A parametric study needs to be conducted in order to optimize the
structural design approach that best takes advantage of the resistance mechanisms
highlighted in this study to achieve the maximum benefits in terms of material resistance and
displacement ductility capacity.
Most of the current nonlinear finite element approaches cannot predict the failure of
the reinforced concrete structures. The main reason is that the modeling is based on the
continuum assumption where cracking is not modeled explicitly. Rather, a failure criteria are
implemented in order to capture failure. There is still a long way for general threedimensional modeling to get reasonable results on stiff reinforced concrete structures with
brittle failure modes due to numerical problems along with the difficulties associated with
capturing and modeling concrete cracking and crushing. Such challenges indicate that
phenomenologically-based models may be a better alternative to continuum and plasticitybased finite element approaches for simulating the behavior of stiff and brittle reinforced
concrete structures.
Concrete damage on structural walls in simplified capacity assessment models was
evaluated with a concrete softening parameter. However, concrete damage due to multidirectional loading failed to be considered. Another parameter that accounts for such loading
effects needs to be included.

209

8.4 Research Impact
This study highlighted the application of high-strength-concrete in seismic regions
and the investigation of the web crushing capacity and failure mechanisms. It has been
demonstrated that HSC can enhance the ductility of structural walls by increasing web
crushing capacity. A new ductile failure concept, namely that of “ductile shear failures” was
proposed and demonstrated to manifest itself with the use of HSC in thin webbed elements
constrained by well confined boundary elements that subject the web to flexure/shear
effects. Such a failure mode, which was demonstrated to be reliably designed for and
predicted for compressive strengths of up to 103 MPa (15 ksi) can be a viable ductile failure
mode for seismic design according to a capacity based design philosophy.
The shear design provisions in current design codes were verified to be over
conservative. This study provides valuable information on the extension of the current codes
to high-strength-concrete. The simplified analytical method proposed in this study can be
used to evaluate the web crushing capacity of high-strength-concrete structural walls and
wall assemblies.

210

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211

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